# Working Papers Series

The Working Paper Series are pre-publication versions of research reports associated with the Centre's staff, research students and visitors.**Links to abstracts and downloadable pdf versions of the papers can be found below:**

## Research Papers

### 2021

#### Scaling properties of a moving polymer

**Mueller C. and Neuman, E.**

We set up an SPDE model for a moving, weakly self-avoiding polymer with intrinsic length *J* taking values in (0,∞). Our main result states that the effective radius of the polymer is approximately *J ^{5/3}*; evidently for large

*J*the polymer undergoes stretching. This contrasts with the equilibrium situation without the time variable, where many earlier results show that the effective radius is approximately

*J*.

For such a moving polymer taking values in **R**^{2}, we offer a con- jecture that the effective radius is approximately *J ^{5/4}*.

#### The multiplicative chaos of *H*=0 fractional Brownian fields

**Hager P. and Neuman E.**

Abstract: please view the abstract here.

#### Price Impact on Term Structure

**Brigo D., Graceffa F. and Neuman E.**

We introduce a first theory of price impact in presence of an interest-rates term structure. We explain how one can formulate instantaneous and transient price impact on bonds with different maturities, including a cross price impact that is endogenous to the term structure. We connect the introduced impact to classic no-arbitrage theory for interest rate markets, showing that impact can be embedded in the pricing measure and that no-arbitrage can be preserved. We present pricing examples in presence of price impact and numerical examples of how impact changes the shape of the term structure. Finally, to show that our approach is applicable we solve an optimal execution problem in interest rate markets with the type of price impact we developed in the paper.

#### The Log Moment formula for implied volatility

**Jacquier, A. and Raval, V.**

We revisit the foundational Moment Formula proved by Roger Lee fifteen years ago. We show that when the underlying stock price martingale admits finite log-moments E[|log(S)|^q] for some positive q, the arbitrage-free growth in the left wing of the implied volatility smile is less constrained than Lee's bound. The result is rationalised by a market trading discretely monitored variance swaps wherein the payoff is a function of squared log-returns, and requires no assumption for the underlying martingale to admit any negative moment. In this respect, the result can derived from a model-independent setup. As a byproduct, we relax the moment assumptions on the stock price to provide a new proof of the notorious Gatheral-Fukasawa formula expressing variance swaps in terms of the implied volatility.

#### Large and moderate deviations for stochastic Volterra systems

**Jacquier, A. and Pannier, A.**

We provide a unified treatment of pathwise Large and Moderate deviations principles for a general class of multidimensional stochastic Volterra equations with singular kernels, not necessarily of convolution form. Our methodology is based on the weak convergence approach by Budhijara, Dupuis and Ellis. We show in particular how this framework encompasses most rough volatility models used in mathematical finance and generalises many recent results in the literature.

#### An Equilibrium Model for the Cross-Section of Liquidity Premia

**Muhle-Karbe, J., Shi, X., Yang, C.**

We study a risk-sharing economy where an arbitrary number of heterogenous agents trades an arbitrary number of risky assets subject to quadratic transaction costs. For linear state dynamics, the forward-backward stochastic differential equations characterizing equilibrium asset prices and trading strategies in this context reduce to a system of matrix-valued Riccati equations. We prove the existence of a unique global solution and provide explicit asymptotic expansions that allow us to approximate the corresponding equilibrium for small transaction costs. These tractable approximation formulas make it feasible to calibrate the model to time series of prices and trading volume, and to study the cross-section of liquidity premia earned by assets with higher and lower trading costs. This is illustrated by an empirical case study.

#### Functional quantization of rough volatility and applications to the VIX

**Bonesini, O., Callegaro, G., Jacquier, A.**

We develop a product functional quantization of rough volatility. Since the quantizers can be computed offline, this new technique, built on the insightful works by Luschgy and Pages, becomes a strong competitor in the new arena of numerical tools for rough volatility. We concentrate our numerical analysis to pricing VIX Futures in the rough Bergomi model and compare our results to other recently suggested benchmarks.

#### Skorohod and rough integration for stochastic differential equations driven by Volterra processes

**Cass, T., and Lim, N.**

Given a solution Y to a rough differential equation (RDE), a recent result [7] extends the classical Ito-Stratonovich formula and provides a closed-form expression for ∫ Y ○ dX − ∫ Y dX, i.e. the difference between the rough and Skorohod integrals of Y with respect to X, where X is a Gaussian process with finite p-variation less than 3. In this paper, we extend this result to Gaussian processes with finite p-variation such that 3 ≤ p < 4. The constraint this time is that we restrict ourselves to Volterra Gaussian processes with kernels satisfying a natural condition, which however still allows the result to encompass many standard examples, including fractional Brownian motion with Hurst parameter H > 1/4. As an application we recover Ito formulas in the case where the vector fields of the RDE governing Y are commutative.

#### Long-time behaviour of degenerate diffusions: UFG-type SDEs and time-inhomogeneous hypoelliptic processes

**Cass, T., Crisan, D., Dobson, P., Ottobre, M. **

We study the long time behaviour of a large class of diffusion processes onRN,generated by second order differential operators of (possibly) degenerate type. Theoperators that we consider need not satisfy the Hörmander Condition (HC). Instead,they satisfy the so-called UFG condition, introduced by Herman, Lobry and Sussmanin the context of geometric control theory and later by Kusuoka and Stroock. Wedemonstrate the importance of the class of UFG processes in several respects: i) weshow that UFG processes constitute a family of SDEs which exhibit, in general,multiple invariant measures (i.e. they are in general non-ergodic) and for which oneis able to describe a systematic procedure to study the basin of attraction of eachinvariant measure (equilibrium state). ii) We use an explicit change of coordinatesto prove that every UFG diffusion can be, at least locally, represented as a systemconsisting of an SDE coupled with an ODE, where the ODE evolves independentlyof the SDE part of the dynamics. iii) As a result, UFG diffusions are inherently “lesssmooth" than hypoelliptic SDEs; more precisely, we prove that UFG processes do notadmit a density with respect to Lebesgue measure on the entire space, but only onsuitable time-evolving submanifolds, which we describe. iv) We show that our resultsand techniques, which we devised for UFG processes, can be applied to the study ofthe long-time behaviour of non-autonomous hypoelliptic SDEs and therefore produceseveral results on this latter class of processes as well. v) Because processes thatsatisfy the (uniform) parabolic HC are UFG processes, this paper contains a wealth ofresults about the long time behaviour of (uniformly) hypoelliptic processes which arenon-ergodic.

#### Option pricing models without probability: A rough paths approach

**Armstrong, J., Bellani, C., Brigo, D., Cass, T.**

We describe the pricing and hedging of financial options without the use ofprobability using rough paths. By encoding the volatility of assets in anenhancement of the price trajectory, we give a pathwise presentation of thereplication of European options. The continuity properties of rough-paths allowus to generalise the so-called fundamental theorem of derivative trading,showing that a small misspecification of the model will yield only a smallexcess profit or loss of the replication strategy. Our hedging strategy is anenhanced version of classical delta hedging where we use volatility swaps tohedge the second order terms arising in rough-path integrals, resulting inimproved robustness.

#### The Signature Kernel is the solution of a Goursat PDE

**Salvi, C., Cass, T., Foster, J., Lyons, T., Yang, W. **

Recently, there has been an increased interest in the development of kernel methods for learning with sequential data. The signature kernel is a learning tool with potential to handle irregularly sampled, multivariate time series. In "Kernels for sequentially ordered data" the authors introduced a kernel trick for the truncated version of this kernel avoiding the exponential complexity that would have been involved in a direct computation. Here we show that for continuously differentiable paths, the signature kernel solves a hyperbolic PDE and recognize the connection with a well known class of differential equations known in the literature as Goursat problems. This Goursat PDE only depends on the increments of the input sequences, does not require the explicit computation of signatures and can be solved efficiently using state-of-the-arthyperbolic PDE numerical solvers, giving a kernel trick for the untruncated signature kernel, with the same raw complexity as the method from "Kernels for sequentially ordered data", but with the advantage that the PDE numerical scheme is well suited for GPU parallelization, which effectively reduces the complexity by a full order of magnitude in the length of the input sequences. In addition, we extend the previous analysis to the space of geometric rough paths and establish, using classical results from rough path theory, that the rough version of the signature kernel solves a rough integral equation analogous to the aforementioned Goursat PDE. Finally, we empirically demonstrate the effectiveness of our PDE kernel as a machine learning tool in various machine learning applications dealing with sequential data. We release the library sigkernel publicly available at this https URL.

#### A combinatorial approach to geometric rough paths and their controlled paths

**Cass, T., Driver, B., Litterer, C., Ferrucci, E.**

Preprint, available on request.

#### Deep Hedging: Learning Risk-Neutral Implied Volatility Dynamics

**Buehler, H., Murray, P., Pakkanen, M., Wood, B.**

We present a numerically efficient approach for learning a risk-neutral measure for paths of simulated spot and option prices up to a finite horizon under convex transaction costs and convex trading constraints. This approach can then be used to implement a stochastic implied volatility model in the following two steps: 1. Train a market simulator for option prices, as discussed for example in our recent; 2. Find a risk-neutral density, specifically the minimal entropy martingale measure. The resulting model can be used for risk-neutral pricing, or for Deep Hedging in the case of transaction costs or trading constraints. To motivate the proposed approach, we also show that market dynamics are free from "statistical arbitrage" in the absence of transaction costs if and only if they follow a risk-neutral measure. We additionally provide a more general characterization in the presence of convex transaction costs and trading constraints. These results can be seen as an analogue of the fundamental theorem of asset pricing for statistical arbitrage under trading frictions and are of independent interest.

#### Score-Based Parameter Estimation for a Class of Continuous-Time State Space Models

**Beskos, A., Crisan, D., Jasra, A., Kantas, N., Ruzayqat, H.**

We consider the problem of parameter estimation for a class of continuous-time state space models. In particular, we explore the case of a partially observed diffusion, with data also arriving according to a diffusion process. Based upon a standard identity of the score function, we consider two particle filter based methodologies to estimate the score function. Both methods rely on an online estimation algorithm for the score function of O(N2) cost, with N∈N the number of particles. The first approach employs a simple Euler discretization and standard particle smoothers and is of cost O(N2+NΔ−1l) per unit time, where Δl=2−l, l∈N0, is the time-discretization step. The second approach is new and based upon a novel diffusion bridge construction. It yields a new backward type Feynman-Kac formula in continuous-time for the score function and is presented along with a particle method for its approximation. Considering a time-discretization, the cost is O(N2Δ−1l) per unit time. To improve computational costs, we then consider multilevel methodologies for the score function. We illustrate our parameter estimation method via stochastic gradient approaches in several numerical examples.

#### On the radius of Gaussian free field excursion clusters

**Goswami, S., Rodriguez, P., Severo, F.**

We investigate the Gaussian free field φ on Zd, for d ≥ 3, and give sharp bounds on the probability that the radius of a finite cluster in the excursion set {φ ≥ h} exceeds a large value N, for any height h not equal to h_*, where h_* refers to the corresponding percolation critical parameter. In dimension d = 3, we prove that this probability is sub-exponential in N and decays as exp\{-(π\/6) (h-h_* )^2 N\/ logN \} as N→∞ to principal exponential order. When d ≥ 4, we prove that these tails decay exponentially in N. Our results extend to other quantities of interest, such as truncated two-point functions and the two-arms probability for annuli crossings at scale N.

#### Cluster capacity functionals and isomorphism theorems for Gaussian free fields

**Drewitz, A., Prévost, A., Rodriguez, P.**

We investigate level sets of the Gaussian free field on continuous transient metric graphs G ̃ and study the capacity of its level set clusters. We prove, without any further assumption on the base graph G that the capacity of sign clusters on G ̃ is finite almost surely. This leads to a new and effective criterion to determine whether the sign clusters of the free field on G ̃ are bounded or not. It also elucidates why the critical parameter for percolation of level sets on G ̃ vanishes in most instances in the massless case and establishes the continuity of this phase transition in a wide range of cases, including all vertex-transitive graphs. When the sign clusters on G ̃ do not percolate, we further determine by means of isomorphism theory the exact law of the capacity of compact clusters at any height. Specifically, we derive this law from an extension of Sznitman's refinement of Lupu's recent isomorphism theorem relating the free field and random interlacements, proved along the way, and which holds under the sole assumption that sign clusters on G ̃ are bounded. Finally, we show that the law of the cluster capacity functionals obtained in this way actually characterizes the isomorphism theorem, i.e. the two are equivalent.

#### Critical exponents for a percolation model on transient graphs

**Drewitz, A., Prévost, A., Rodriguez, P.**

We consider the bond percolation problem on a transient weighted graph induced by the excursion sets of the Gaussian free field on the corresponding cable system. Owing to the continuity of this setup and the strong Markov property of the field on the one hand, and the links with potential theory for the associated diffusion on the other, we rigorously determine the behavior of various key quantities related to the (near-)critical regime for this model. In particular, our results apply in case the base graph is the three-dimensional cubic lattice. They unveil the values of the associated critical exponents, which are explicit but not mean-field and consistent with predictions from scaling theory below the upper-critical dimension.

#### Contagious McKean-Vlasov Systems with heterogeneity

**Feinstein, Z., Sojmark, A.**

We introduce a heterogeneous version of a contagious McKean-Vlasov system whose heterogeneity has a natural and particularly tractable structure. It is shown that this system characterises the limit points of a finite particle system with applications to solvency contagion in interbank markets and coupled integrate-and-fire models in mathematical neuroscience. Furthermore, we provide a result on global uniqueness for the full common noise system under a simple smallness condition on the strength of interactions, and we show that there is a unique differentiable solution up to an explosion time in the system without common noise.

#### Functional Limit Theorems for Volterra Processes and Applications to Homogenization

**Johann Gehringer, Xue-Mei Li, Julian Sieber**

We prove an enhanced limit theorem for additive functionals of a multidimensional Volterra process (yt)t≥0. As an application, we establish weak convergence of the solutions of rough differential equations (RDE) of the form dxεt=1ε√f(xεt,ytε)dt+g(xεt)dBt, and identify their limits as solutions of an RDE driven by a Gaussian field with a drift coming from the Lévy area correction of the limiting rough driver. The equation models a passive tracer in a random field.

In particular if h is random field such that h(x,⋅) a semi-martingale with spatial parameter x, we show that the solutions of the equations dxεt=1ϵ√f(xεt,ytε)dt+h(xεt,dt), converge weakly to that of a Kunita type Itô SDE dxt=G(xt,dt) where G(x,t) is a semi-martingale with spatial parameters. Furthermore the N-point motions converge.

### 2020

**Semi-martingale driven variational principles**

**Crisan, D., Street, O.**

Spearheaded by the recent efforts to derive stochastic geophysical fluid dynamics models, we present a generic framework for introducing stochasticity into variational principles through the concept of a semi-martingale driven variational principle and constraining the component variables to be compatible with the driving semi-martingale. Within this framework and the corresponding choice of constraints, the Euler-Poincare equation can be easily deduced. We show that their corresponding deterministic counterparts are particular cases of this class of stochastic variational principles. Moreover, this is a natural framework that enables us to correctly characterize the pressure term in incompressible stochastic fluid models. Other general constraints can also be incorporated as long as they are compatible with the driving semi-martingale.

**Optimal Signal-Adaptive Trading with Temporary and Transient Price Impact**

**Eyal Neuman, Moritz Voß**

We study optimal liquidation in the presence of linear temporary and transient price impact along with taking into account a general price predicting finite-variation signal. We formulate this problem as minimization of a cost-risk functional over a class of absolutely continuous and signal-adaptive strategies. The stochastic control problem is solved by following a probabilistic and convex analytic approach. We show that the optimal trading strategy is given by a system of four coupled forward-backward SDEs, which can be solved explicitly. Our results reveal how the induced transient price distortion provides together with the predictive signal an additional predictor about future price changes. As a consequence, the optimal signal-adaptive trading rate trades off exploiting the predictive signal against incurring the transient displacement of the execution price from its unaffected level. This answers an open question from Lehalle and Neuman [27] as we show how to derive the unique optimal signal-adaptive liquidation strategy when price impact is not only temporary but also transient.

**Path-dependent volatility models**

**Antoine Jacquier, Chloe Lacombe**

We provide a thorough analysis of the path-dependent volatility model introduced by Guyon, proving existence and uniqueness of a strong solution, characterising its behaviour at boundary points, and deriving large deviations estimates. We further develop a numerical algorithm in order to jointly calibrate SP500 and VIX market data.

**Well-Posedness and Equilibrium Behaviour of Overdamped Dynamics Density Functional Theory**

**B. D. Goddard, R. D. Mills-Williams, Grigorios Pavliotis**

We establish the global well-posedness of overdamped dynamical density functional theory (DDFT): a nonlinear, nonlocal integro-partial differential equation used in statistical mechanical models of colloidal flow and other applications including nonlinear reaction-diffusion systems and opinion dynamics. With no-flux boundary conditions, we determine the well-posedness of the full nonlocal equations including two-body hydrodynamic interactions (HI) through the theory of Fredolm operators. Principally, this is done by rewriting the dynamics for the density ϱ as a nonlocal Smoluchowski equation with a non-constant diffusion tensor D dependent on the diagonal part (Z1) of the HI tensor, and an effective drift A[a] dependent on the off-diagonal part (Z2). We derive a scheme to uniquely construct the mean colloid flux a(r,t) in terms of eigenvectors of D, show that the stationary density ϱ(r) is independent of the HI tensors, as well as proving exponentially fast convergence to equilibrium. The stability of the equilibria ϱ(r) is studied by considering the bounded (nonlocal) perturbation of the differential (local) part of the linearised operator. We show that the spectral properties of the full nonlocal operator with no-flux boundary conditions can differ considerably from those with periodic boundary conditions. We showcase our results by using the numerical methods available in the pseudo-spectral collocation scheme 2DChebClass.

**On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions**

**Matias G. Delgadino, Rishabh S. Gvalani, Grigorios A. Pavliotis**

The objective of this article is to analyse the statistical behaviour of a large number of weakly interacting diffusion processes evolving under the influence of a periodic interaction potential. We focus our attention on the combined mean field and diffusive(homogenisation)limits. In particular, we show that these two limits do not commute if the mean field system constrained to the torus undergoes a phase transition, that is to say if it admits more than one steady state. A typical example of such a system on the torus is given by the noisy Kuramoto model of mean field plane rotators. As a by-product of our main results, we also analyse the energetic consequences of the central limit theorem for fluctuations around the mean field limit and derive optimal rates of convergence in relative entropy of the Gibbs measure to the (unique) limit of the mean field energy below the critical temperature.

**Non-Geometric Rough Paths on Manifolds**

**John Armstrong, Damiano Brigo, Thomas Cass, Emilio Rossi Ferrucci**

We provide a theory of manifold-valued rough paths of bounded 3 > p-variation, which we do not assume to be geometric. Rough paths are defined in charts, and coordinate-free (but connection-dependent) definitions of the rough integral of cotangent bundle-valued controlled paths, and of RDEs driven by a rough path valued in another manifold, are given. When the path is the realisation of semimartingale we recover the theory of Itô integration and SDEs on manifolds [É89]. We proceed to present the extrinsic counterparts to our local formulae, and show how these extend the work in [CDL15] to the setting of non-geometric rough paths and controlled integrands more general than 1-forms. In the last section we turn to parallel transport and Cartan development: the lack of geometricity leads us to make the choice of a connection on the tangent bundle of the manifold TM, which figures in an Itô correction term in the parallelism RDE; such connection, which is not needed in the geometric/Stratonovich setting, is required to satisfy properties which guarantee well-definedness, linearity, and optionally isometricity of parallel transport. We conclude by providing numerous examples, some accompanied by numerical simulations, which explore the additional subtleties introduced by our change in perspective.

**Option pricing models without probability: a rough paths approach**

**John Armstrong, Claudio Bellani, Damiano Brigo, Thomas Cass**

We describe the pricing and hedging of financial options without the use of probability using rough paths. By encoding the volatility of assets in an enhancement of the price trajectory, we give a pathwise presentation of the replication of European options. The continuity properties of rough-paths allow us to generalise the so-called fundamental theorem of derivative trading, showing that a small misspecification of the model will yield only a small excess profit or loss of the replication strategy. Our hedging strategy is an enhanced version of classical delta hedging where we use volatility swaps to hedge the second order terms arising in rough-path integrals, resulting in improved robustness.

#### Long-time behaviour of degenerate diffusions: UFG-type SDEs and time-inhomogeneous hypoelliptic processes

**T. Cass, D. Crisan, P. Dobson, M. Ottobre**

We study the long time behaviour of a large class of diffusion processes on RN, generated by second order differential operators of (possibly) degenerate type. The operators that we consider {\em need not} satisfy the Hörmander condition. Instead, they satisfy the so-called UFG condition, introduced by Herman, Lobry and Sussman in the context of geometric control theory and later by Kusuoka and Stroock, this time with probabilistic motivations. In this paper we study UFG diffusions and demonstrate the importance of such a class of processes in several respects: roughly speaking i) we show that UFG processes constitute a family of SDEs which exhibit multiple invariant measures and for which one is able to describe a systematic procedure to determine the basin of attraction of each invariant measure (equilibrium state). ii) We use an explicit change of coordinates to prove that every UFG diffusion can be, at least locally, represented as a system consisting of an SDE coupled with an ODE, where the ODE evolves independently of the SDE part of the dynamics. iii) As a result, UFG diffusions are inherently "less smooth" than hypoelliptic SDEs; more precisely, we prove that UFG processes do not admit a density with respect to Lebesgue measure on the entire space, but only on suitable time-evolving submanifolds, which we describe. iv) We show that our results and techniques, which we devised for UFG processes, can be applied to the study of the long-time behaviour of non-autonomous hypoelliptic SDEs and therefore produce several results on this latter class of processes as well. v) Because processes that satisfy the (uniform) parabolic Hörmander condition are UFG processes, our paper contains a wealth of results about the long time behaviour of (uniformly) hypoelliptic processes which are non-ergodic, in the sense that they exhibit multiple invariant measures.

#### Rough functional quantization and the support of McKean-Vlasov equations

**Thomas Cass, Goncalo dos Reis, William Salkeld**

We prove a representation for the support of McKean Vlasov Equations. To do so, we construct functional quantizations for the law of Brownian motion as a measure over the (non-reflexive) Banach space of Hölder continuous paths. By solving optimal Karhunen Loève expansions and exploiting the compact embedding of Gaussian measures, we obtain a sequence of deterministic finite supported measures that converge to the law of a Brownian motion with explicit rate. We show the approximation sequence is near optimal with very favourable integrability properties and prove these approximations remain true when the paths are enhanced to rough paths. These results are of independent interest.

The functional quantization results then yield a novel way to build deterministic, finite supported measures that approximate the law of the McKean Vlasov Equation driven by the Brownian motion which crucially avoid the use of random empirical distributions. These are then used to solve an approximate skeleton process that characterises the support of the McKean Vlasov Equation.

We give explicit rates of convergence for the deterministic finite supported measures in rough-path Hölder metrics and determine the size of the particle system required to accurately estimate the law of McKean Vlasov equations with respect to the Hölder norm.

#### Computing the untruncated signature kernel as the solution of a Goursat problem

**Cass, T., Lyons, T., Yang, W., Salvi, C.**

Recently there has been an increased interest in the development of kernel methods for learning with sequential data. The truncated signature kernel is a new learning tool designed to handle irregularly sampled, multidimensional data streams. In this article we consider the untruncated signature kernel and show that for paths of bounded variation it is the solution of a Goursat problem. This linear hyperbolic PDE only depends on the increments of the input sequences, doesn't require the explicit computation of signatures and can be solved using any PDE numerical solver; it is a kernel trick for the untruncated signature kernel. In addition, we extend the analysis to the space of geometric rough paths, and establish using classical results from stochastic analysis that the rough version of the untruncated signature kernel solves a rough integral equation analogous to the Goursat problem for the bounded variation case. Finally we empirically demonstrate the effectiveness of this kernel in two data science applications: multivariate time-series classification and dimensionality reduction.

#### Mean field limits for interacting diffusions with colored noise: phase transitions and spectral numerical methods

**Gomes, S., Pavliotis, G., Vaes, U.**

In this paper we consider systems of weakly interacting particles driven by colored noise in a bistable potential, and we study the effect of the correlation time of the noise on the bifurcation diagram for the equilibrium states. We accomplish this by solving the corresponding McKean--Vlasov equation using a Hermite spectral method, and we verify our findings using Monte Carlo simulations of the particle system. We consider both Gaussian and non-Gaussian noise processes, and for each model of the noise we also study the behavior of the system in the small correlation time regime using perturbation theory. The spectral method that we develop in this paper can be used for solving linear and nonlinear, local and nonlocal (mean field) Fokker--Planck equations, without requiring that they have a gradient structure.

#### Long-time behaviour and phase transitions for the McKean-Vlasov equation on the torus

**J Carrillo, R Gvalani, G Pavliotis, A Schlichting**

We study the McKean–Vlasov equation ∂t = β−1 + κ ∇·(∇(W )), with periodic boundary conditions on the torus. We first study the global asymptotic stability of the homogeneous steady state. We then focus our attention on the stationary system, and prove the existence of nontrivial solutions branching from the homogeneous steady state, through possibly infinitely many bifurcations, under appropriate assumptions on the interaction potential. We also provide sufficient conditions for the existence of continuous and discontinuous phase transitions. Finally, we showcase these results by applying them to several examples of interaction potentials such as the noisy Kuramoto model for synchronisation, the Keller–Segel model for bacterial chemotaxis, and the noisy Hegselmann–Krausse model for opinion dynamics.

#### At the mercy of the common noise: blow-ups in a conditional McKean-Vlasov problem

**A Sojmark, S Ledger**

We consider a model of positive feedback and contagion in large mean-field systems, affected by a common source of noise driven by Brownian motion. Although the driving dynamics are continuous, the positive feedback effect can lead to `blow-up' phenomena whereby solutions develop jump-discontinuities. Our main results are twofold and concern the conditional McKean--Vlasov formulation of the model. First and foremost, we show that there are global solutions to this McKean--Vlasov problem, which can be realised as limit points of a motivating particle system with common noise. Furthermore, we derive results on the occurrence of blow-ups, thereby showing how these events can be triggered or prevented by the pathwise realisations of the common noise. This preprint is a much extended version of an earlier preprint from 2018, rewritten in its entirety and presenting new results on blow-ups.

#### Dynamic default contagion in interbank systems

**A Sojmark, Z Feinstein**

In this work we provide a simple setting that connects the structural modelling approach of Gai-Kapadia interbank networks with the mean-field approach to default contagion. To accomplish this we make two key contributions. First, we propose a dynamic default contagion model with endogenous early defaults for a finite set of banks, generalising the Gai-Kapadia framework. Second, we reformulate this system as a stochastic particle system leading to a limiting mean-field problem. We study the existence of these clearing systems and, for the mean-field problem, the continuity of the system response.

#### Integration by parts formulae for the laws of Bessel bridges via hypergeometric functions

**Henri Elad Altman**

In this article, we extend the integration by parts formulae (IbPF) for the laws of Bessel bridges obtained in a recent work by Elad Altman and Zambotti to linear functionals. Our proof relies on properties of hypergeometric functions, thus providing a new interpretation of these formulae.

#### Roughness in spot variance? A GMM approach for estimation of fractional log-normal stochastic volatility models using realized measures

**Anine E. Bolko, Kim Christensen, Mikko S. Pakkanen and Bezirgen Veliyev**

In this paper, we develop a generalized method of moments approach for joint estimation of the parameters of a fractional log-normal stochastic volatility model. We show that with an arbitrary Hurst exponent an estimator based on integrated variance is consistent. Moreover, under stronger conditions we also derive a central limit theorem. These results stand even when integrated variance is replaced with a realized measure of volatility calculated from discrete high-frequency data. However, in practice a realized estimator contains sampling error, the effect of which is to skew the fractal coefficient toward "roughness". We construct an analytical approach to control this error. In a simulation study, we demonstrate convincing small sample properties of our approach based both on integrated and realized variance over the entire memory spectrum. We show that the bias correction attenuates any systematic deviance in the estimated parameters. Our procedure is applied to empirical high-frequency data from numerous leading equity indexes. With our robust approach the Hurst index is estimated around 0.05, confirming roughness in integrated variance.

#### Limit theorems for trawl processes

**Mikko S. Pakkanen, Riccardo Passeggeri, Orimar Sauri and Almut E. D. Veraart**

In this work we derive limit theorems for trawl processes. First,we study the asymptotic behaviour of the partial sums of the discretized trawl process (X_{iΔn})^{⌊}^{nt}^{⌋}^{−1}_{i=0}, under the assumption that as n↑∞, Δ_{n}↓0 and nΔ_{n}→μ∈[0,+∞]. Second, we derive a functional limit theorem for trawl processes as the Lévy measure of the trawl seed grows to infinity and show that the limiting process has a Gaussian moving average representation.

#### Non-Geometric Rough Paths on Manifolds

**Armstrong, J., Brigo, D., Cass, T., Rossi Ferrucci, E. **

We provide a theory of manifold-valued rough paths of bounded 3 > p-variation, which we do not assume to be geometric. Rough paths are defined in charts, and coordinate-free (but connection-dependent) definitions of the rough integral of cotangent bundle-valued controlled paths, and of RDEs driven by a rough path valued in another manifold, are given. When the path is the realisation of semimartingale we recover the theory of Itô integration and SDEs on manifolds [É89]. We proceed to present the extrinsic counterparts to our local formulae, and show how these extend the work in [CDL15] to the setting of non-geometric rough paths and controlled integrands more general than 1-forms. In the last section we turn to parallel transport and Cartan development: the lack of geometricity leads us to make the choice of a connection on the tangent bundle of the manifold TM, which figures in an Itô correction term in the parallelism RDE; such connection, which is not needed in the geometric/Stratonovich setting, is required to satisfy properties which guarantee well-definedness, linearity, and optionally isometricity of parallel transport. We conclude by providing numerous examples, some accompanied by numerical simulations, which explore the additional subtleties introduced by our change in perspective.

#### Feasible Inference for Stochastic Volatility in Brownian Semistationary Processes

**Murray, P., Passeggeri, R., Veraart, A., Pakkanen, M.**

This article studies the finite sample behaviour of a number of estimators for the integrated power volatility process of a Brownian semistationary process in the non semi-martingale setting. We establish three consistent feasible estimators for the integrated volatility, two derived from parametric methods and one non-parametrically. We then use a simulation study to compare the convergence properties of the estimators to one another, and to a benchmark of an infeasible estimator. We further establish bounds for the asymptotic variance of the infeasible estimator and assess whether a central limit theorem which holds for the infeasible estimator can be translated into a feasible limit theorem for the non-parametric estimator.

#### Limit Theorems for Trawl Processes

**Pakkanen, M., Passeggeri, R., Sauri, O., Veraart, A.**

In this work we derive limit theorems for trawl processes. First, we study the asymptotic behaviour of the partial sums of the discretized trawl process (XiΔ_{n})⌊nt⌋−1i=0, under the assumption that as n↑∞, Δ_{n}↓0 and nΔ_{n}→μ∈[0,+∞]. Second, we derive a functional limit theorem for trawl processes as the Lévy measure of the trawl seed grows to infinity and show that the limiting process has a Gaussian moving average representation.

#### Roughness in spot variance? A GMM approach for estimation of fractional log-normal stochastic volatility models using realized measures

**Bolko, A., Christensen, K., Pakkanen, M., Veliyev, B.**

In this paper, we develop a generalized method of moments approach for joint estimation of the parameters of a fractional log-normal stochastic volatility model. We show that with an arbitrary Hurst exponent an estimator based on integrated variance is consistent. Moreover, under stronger conditions we also derive a central limit theorem. These results stand even when integrated variance is replaced with a realized measure of volatility calculated from discrete high-frequency data. However, in practice a realized estimator contains sampling error, the effect of which is to skew the fractal coefficient toward "roughness". We construct an analytical approach to control this error. In a simulation study, we demonstrate convincing small sample properties of our approach based both on integrated and realized variance over the entire memory spectrum. We show that the bias correction attenuates any systematic deviance in the estimated parameters. Our procedure is applied to empirical high-frequency data from numerous leading equity indexes. With our robust approach the Hurst index is estimated around 0.05, confirming roughness in integrated variance.

#### Two-Timescale Stochastic Gradient Descent in Continuous Time with Applications to Joint Online Parameter Estimation and Optimal Sensor Placement

**Sharrock, L., Kantas, N.**

In this paper, we establish the almost sure convergence of two-timescale stochastic gradient descent algorithms in continuous time under general noise and stability conditions, extending well known results in discrete time. We analyse algorithms with additive noise and those with non-additive noise. In the non-additive case, our analysis is carried out under the assumption that the noise is a continuous-time Markov process, controlled by the algorithm states. The algorithms we consider can be applied to a broad class of bilevel optimisation problems. We study one such problem in detail, namely, the problem of joint online parameter estimation and optimal sensor placement for a partially observed diffusion process. We demonstrate how this can be formulated as a bilevel optimisation problem, and propose a solution in the form of a continuous-time, two-timescale, stochastic gradient descent algorithm. Furthermore, under suitable conditions on the latent signal, the filter, and the filter derivatives, we establish almost sure convergence of the online parameter estimates and optimal sensor placements to the stationary points of the asymptotic log-likelihood and asymptotic filter covariance, respectively. We also provide numerical examples, illustrating the application of the proposed methodology to a partially observed Beneš equation, and a partially observed stochastic advection-diffusion equation.

#### On stochastic mirror descent with interacting particles: convergence properties and variance reduction

**Borovykh, A., Kantas, N., Parpas, P., Pavliotis, G.**

An open problem in optimization with noisy information is the computation of an exact minimizer that is independent of the amount of noise. A standard practice in stochastic approximation algorithms is to use a decreasing step-size. This however leads to a slower convergence. A second alternative is to use a fixed step-size and run independent replicas of the algorithm and average these. A third option is to run replicas of the algorithm and allow them to interact. It is unclear which of these options works best. To address this question, we reduce the problem of the computation of an exact minimizer with noisy gradient information to the study of stochastic mirror descent with interacting particles. We study the convergence of stochastic mirror descent and make explicit the tradeoffs between communication and variance reduction. We provide theoretical and numerical evidence to suggest that interaction helps to improve convergence and reduce the variance of the estimate.

#### On the Generalised Langevin Equation for Simulated Annealing

**Chak, M., Kantas, N., Pavliotis, G.**

In this paper, we consider the generalised (higher order) Langevin equation for the purpose of simulated annealing and optimisation of nonconvex functions. Our approach modifies the underdamped Langevin equation by replacing the Brownian noise with an appropriate Ornstein-Uhlenbeck process to account for memory in the system. Under reasonable conditions on the loss function and the annealing schedule, we establish convergence of the continuous time dynamics to a global minimum. In addition, we investigate the performance numerically and show better performance and higher exploration of the state space compared to the underdamped and overdamped Langevin dynamics with the same annealing schedule.

#### Joint Online Parameter Estimation and Optimal Sensor Placement for the Partially Observed Stochastic Advection-Diffusion Equation

**Sharrock, L., Kantas, N.**

In this paper, we consider the problem of jointly performing online parameter estimation and optimal sensor placement for a partially observed infinite dimensional linear diffusion process. We present a novel solution to this problem in the form of a continuous-time, two-timescale stochastic gradient descent algorithm, which recursively seeks to maximise the log-likelihood with respect to the unknown model parameters, and to minimise the expected mean squared error of the hidden state estimate with respect to the sensor locations. We also provide extensive numerical results illustrating the performance of the proposed approach in the case that the hidden signal is governed by the two-dimensional stochastic advection-diffusion equation.

#### Equality of critical parameters for percolation of Gaussian free field level-sets

**Duminil-Copin, H., Goswami, S., Rodriguez, P., Severo, F.**

We consider level-sets of the Gaussian free field on Zd, for d ≥ 3, above a given real-valued height parameter h. As h varies, this defines a canonical percolation model with strong, algebraically decaying correlations. We prove that three natural critical parameters associated to this model, namely h_(**)(d), h_*(d) and h ̅(d), respectively describing a well-ordered subcritical phase, the emergence of an infinite cluster, and the onset of a local uniqueness regime in the supercritical phase, actually coincide, i.e. h_(**)(d) = h_*(d) = h ̅(d) for any d ≥ 3. At the core of our proof lies a new interpolation scheme aimed at integrating out the long-range dependence of the Gaussian free field. The successful implementation of this strategy relies extensively on certain novel renormalization techniques, in particular to control large-field effects. This approach opens the way to a complete understanding of the off-critical phases of strongly correlated percolation models.

#### A priori bounds for rough differential equations with a non-linear damping term

**Bonnefoi, T., Chandra, A., Moinat, A., Weber, H.**

Abstract: please view the abstract here.

#### Phase transitions for ϕ43

**Chandra, A., Gunaratnam, T., Weber, H.**

We establish a surface order large deviation estimate for the magnetisation of low temperature ϕ43. As a byproduct, we obtain a decay of spectral gap for its Glauber dynamics given by the ϕ43 singular stochastic PDE. Our main technical contributions are contour bounds for ϕ43, which extends 2D results by Glimm, Jaffe, and Spencer (1975). We adapt an argument by Bodineau, Velenik, and Ioffe (2000) to use these contour bounds to study phase segregation. The main challenge to obtain the contour bounds is to handle the ultraviolet divergences of ϕ43 whilst preserving the structure of the low temperature potential. To do this, we build on the variational approach to ultraviolet stability for ϕ43 developed recently by Barashkov and Gubinelli (2019).

#### Langevin dynamic for the 2D Yang-Mills measure

**Chandra, A., Chevyrev, I., Hairer, M., Shen, H.**

We define a natural state space and Markov process associated to the stochastic Yang-Mills heat flow in two dimensions. To accomplish this we first introduce a space of distributional connections for which holonomies along sufficiently regular curves (Wilson loop observables) and the action of an associated group of gauge transformations are both well-defined and satisfy good continuity properties. The desired state space is obtained as the corresponding space of orbits under this group action and is shown to be a Polish space when equipped with a natural Hausdorff metric. To construct the Markov process we show that the stochastic Yang-Mills heat flow takes values in our space of connections and use the "DeTurck trick" of introducing a time dependent gauge transformation to show invariance, in law, of the solution under gauge transformations. Our main tool for solving for the Yang-Mills heat flow is the theory of regularity structures and along the way we also develop a "basis-free" framework for applying the theory of regularity structures in the context of vector-valued noise - this provides a conceptual framework for interpreting several previous constructions and we expect this framework to be of independent interest.

#### Slow-Fast Systems with Fractional Environment and Dynamics

**Xue-Mei Li, Julian Sieber**

We prove an averaging principle for interacting slow-fast systems driven by independent fractional Brownian motions. The mode of convergence is in Hölder norm in probability. We also establish geometric ergodicity for a class of fractional-driven stochastic differential equations, partially improving a recent result of Panloup and Richard.

#### Functional limit theorems for the fractional Ornstein-Uhlenbeck process

**Johann Gehringer, Xue-Mei Li**

We prove a functional limit theorem for vector-valued functionals of the fractional Ornstein-Uhlenbeck process, providing the foundation for the fluctuation theory of slow/fast systems driven by such a noise. Our main contribution is on the joint convergence to a limit with both Gaussian and non-Gaussian components. This is valid for any L2 functions, whereas for functions with stronger integrability properties the convergence is shown to hold in the Hölder topology. As an application we prove a `rough creation' result, i.e. the weak convergence of a family of random smooth curves to a non-Markovian random process with rough sample paths. This includes the second order problem and the kinetic fractional Brownian motion model.

#### Diffusive and rough homogenisation in fractional noise field

**Johann Gehringer, Xue-Mei Li**

With recently developed tools, we prove a homogenisation theorem for a random ODE with short and long-range dependent fractional noise. The effective dynamics are not necessarily diffusions, they are given by stochastic differential equations driven simultaneously by stochastic processes from both the Gaussian and the non-Gaussian self-similarity universality classes. A key lemma for this is the `lifted' joint functional central and non-central limit theorem in the rough path topology.

#### Rough Homogenisation with Fractional Dynamics

**Johann Gehringer, Xue-Mei Li**

We review recent developments of slow/fast stochastic differential equations, and also present a new result on Diffusion Homogenisation Theory with fractional and non-strong-mixing noise and providing new examples.

The emphasise of the review will be on the recently developed effective dynamic theory for two scale random systems with fractional noise: Stochastic Averaging and `Rough Diffusion Homogenisation Theory'. We also study the geometric models with perturbations to symmetries.

### 2019

**Fluctuations around a homogenised semilinear random PDE** **Martin Hairer, Étienne Pardoux**

**The support of singular stochastic PDEs** **Martin Hairer, Philipp Schönbauer**

We obtain a generalisation of the Stroock-Varadhan support theorem for a large class of systems of subcritical singular stochastic PDEs driven by a noise that is either white or approximately self-similar. The main problem that we face is the presence of renormalisation. In particular, it may happen in general that different renormalisation procedures yield solutions with different supports. One of the main steps in our construction is the identification of a subgroup H of the renormalisation group such that any renormalisation procedure determines a unique coset g∘H. The support of the solution then only depends on this coset and is obtained by taking the closure of all solutions obtained by replacing the driving noises by smooth functions in the equation that is renormalised by some element of g∘H.

One immediate corollary of our results is that the Φ43 measure in finite volume has full support and that the associated Langevin dynamic is exponentially ergodic.

**Averaging dynamics driven by fractional Brownian motion ****Martin Hairer, Xue-Mei Li**

We consider slow / fast systems where the slow system is driven by fractional Brownian motion with Hurst parameter H>12. We show that unlike in the case H=12, convergence to the averaged solution takes place in probability and the limiting process solves the 'naïvely' averaged equation. Our proof strongly relies on the recently obtained stochastic sewing lemma.

**Geometric stochastic heat equations ** **Yvain Bruned, Franck Gabriel, Martin Hairer, Lorenzo Zambotti**

We consider a natural class of Rd-valued one-dimensional stochastic PDEs driven by space-time white noise that is formally invariant under the action of the diffeomorphism group on Rd. This class contains in particular the KPZ equation, the multiplicative stochastic heat equation, the additive stochastic heat equation, and rough Burgers-type equations. We exhibit a one-parameter family of solution theories with the following properties:

- For all SPDEs in our class for which a solution was previously available, every solution in our family coincides with the previously constructed solution, whether that was obtained using Itô calculus (additive and multiplicative stochastic heat equation), rough path theory (rough Burgers-type equations), or the Hopf-Cole transform (KPZ equation).

- Every solution theory is equivariant under the action of the diffeomorphism group, i.e. identities obtained by formal calculations treating the noise as a smooth function are valid.

- Every solution theory satisfies an analogue of Itô's isometry.

- The counterterms leading to our solution theories vanish at points where the equation agrees to leading order with the additive stochastic heat equation.

In particular, points 2 and 3 show that, surprisingly, our solution theories enjoy properties analogous to those holding for both the Stratonovich and Itô interpretations of SDEs simultaneously. For the natural noisy perturbation of the harmonic map flow with values in an arbitrary Riemannian manifold, we show that all these solution theories coincide. In particular, this allows us to conjecturally identify the process associated to the Markov extension of the Dirichlet form corresponding to the L2-gradient flow for the Brownian loop measure.

**Data assimilation for a quasi-geostrophic model with circulation-preserving stochastic transport noise** **C Cotter, D Crisan, D Holm, W Pan, I Shevchenko**

This paper contains the latest installment of the authors' project on developing ensemble based data assimilation methodology for high dimensional fluid dynamics models. The algorithm presented here is a particle filter that combines model reduction, tempering, jittering, and nudging. The methodology is tested on a two-layer quasi-geostrophic model for a β-plane channel flow with O(106) degrees of freedom out of which only a minute fraction are noisily observed. The model is reduced by following the stochastic variational approach for geophysical fluid dynamics introduced in Holm (Proc Roy Soc A, 2015) as a framework for deriving stochastic parametrisations for unresolved scales. The reduction is substantial: the computations are done only for O(104) degrees of freedom. We introduce a stochastic time-stepping scheme for the two-layer model and prove its consistency in time. Then, we analyze the effect of the different procedures (tempering combined with jittering and nudging) on the performance of the data assimilation procedure using the reduced model, as well as how the dimension of the observational data (the number of "weather stations") and the data assimilation step affect the accuracy and uncertainty of the results.

**A Particle Filter for Stochastic Advection by Lie Transport (SALT): A case study for the damped and forced incompressible 2D Euler equation****C Cotter, D Crisan, DD Holm, W Pan**

In this work, we apply a particle filter with three additional procedures (model reduction, tempering and jittering) to a damped and forced incompressible 2D Euler dynamics defined on a simply connected bounded domain. We show that using the combined algorithm, we are able to successfully assimilate data from a reference system state (the ``truth") modelled by a highly resolved numerical solution of the flow that has roughly 3.1×106 degrees of freedom for 10 eddy turnover times, using modest computational hardware. The model reduction is performed through the introduction of a stochastic advection by Lie transport (SALT) model as the signal on a coarser resolution. The SALT approach was introduced as a general theory using a geometric mechanics framework from Holm, Proc. Roy. Soc. A (2015). This work follows on the numerical implementation for SALT presented by Cotter et al, SIAM Multiscale Model. Sim. (2019) for the flow in consideration. The model reduction is substantial: The reduced SALT model has 4.9×104 degrees of freedom. Forecast reliability and estimated asymptotic behaviour of the particle filter are also presented.

**Well-posedness for a stochastic 2D Euler equation with transport noise****D Crisan, O Lang**

We prove the existence of a unique global strong solution for a stochastic two-dimensional Euler vorticity equation for incompressible flows with noise of transport type. In particular, we show that the initial smoothness of the solution is preserved. The arguments are based on approximating the solution of the Euler equation with a family of viscous solutions which is proved to be relatively compact using a tightness criterion by Kurtz.

**Uniform in time estimates for the weak error of the Euler method for SDEs and a Pathwise Approach to Derivative Estimates for Diffusion Semigroups****D Crisan, P Dobson, M Ottobre**

We present a criterion for uniform in time convergence of the weak error of the Euler scheme for Stochastic Differential equations (SDEs). The criterion requires i) exponential decay in time of the space-derivatives of the semigroup associated with the SDE and ii) bounds on (some) moments of the Euler approximation. We show by means of examples (and counterexamples) how both i) and ii) are needed to obtain the desired result. If the weak error converges to zero uniformly in time, then convergence of ergodic averages follows as well. We also show that Lyapunov-type conditions are neither sufficient nor necessary in order for the weak error of the Euler approximation to converge uniformly in time and clarify relations between the validity of Lyapunov conditions, i) and ii).

Conditions for ii) to hold are studied in the literature. Here we produce sufficient conditions for i) to hold. The study of derivative estimates has attracted a lot of attention, however not many results are known in order to guarantee exponentially fast decay of the derivatives. Exponential decay of derivatives typically follows from coercive-type conditions involving the vector fields appearing in the equation and their commutators; here we focus on the case in which such coercive-type conditions are non-uniform in space. To the best of our knowledge, this situation is unexplored in the literature, at least on a systematic level. To obtain results under such space-inhomogeneous conditions we initiate a pathwise approach to the study of derivative estimates for diffusion semigroups and combine this pathwise method with the use of Large Deviation Principles.

**Bessel SPDEs with general Dirichlet boundary conditions****Henri Elad Altman**

We generalise the integration by parts formulae obtained in a recent article with Lorenzo Zambotti to Bessel bridges on [0,1] with arbitrary boundary values, as well as Bessel processes with arbitrary initial conditions. This allows us to write, formally, the corresponding dynamics using renormalised local times, thus extending the Bessel SPDEs of the above mentioned article to general Dirichlet boundary conditions. We also prove a dynamical result for the case of dimension 2, by providing a weak construction of the gradient dynamics corresponding to a 2-dimensional Bessel bridge.

**On the gradient dynamics associated with wetting models****Jean-Dominique Deuschel, Henri Elad Altman, Tal Orenshtein**

We prove a tightness result for the reversible gradient dynamics associated with critical wetting models with a shrinking strip, thus answering a conjecture from a recent work by Deuschel and Orenshtein. We also introduce a continuous critical wetting model defined by the law of a Brownian meander tilted by its local time near the origin, and prove its convergence to the law of a reflecting Brownian motion. We further provide a description of the associated gradient dynamics in terms of an SPDE with reflection and attraction, which we conjecture to converge to a Bessel SPDE as introduced a recent work by Elad Altman and Zambotti.

**On the Maximal Displacement of Near-critical Branching Random Walks****Eyal Neuman, Xinghua Zheng**

We consider a branching random walk on Z started by n particles at the origin, where each particle disperses according to a mean-zero random walk with bounded support and reproduces with mean number of offspring 1+θ/n. For t≥0, we study Mnt, the rightmost position reached by the branching random walk up to generation [nt]. Under certain moment assumptions on the branching law, we prove that Mnt/n−−√ converges weakly to the rightmost support point of the local time of the limiting super-Brownian motion. The convergence result establishes a sharp exponential decay of the tail distribution of Mnt. We also confirm that when θ>0, the support of the branching random walk grows in a linear speed that is identical to that of the limiting super-Brownian motion which was studied by Pinsky in [28]. The rightmost position over all generations, M:=suptMnt, is also shown to converge weakly to that of the limiting super-Brownian motion, whose tail is found to decay like a Gumbel distribution when

**Static vs Adaptive Strategies for Optimal Execution with Signals** **Claudio Bellani, Damiano Brigo, Alex Done, Eyal Neuman**

We compare optimal static and dynamic solutions in trade execution. An optimal trade execution problem is considered where a trader is looking at a short-term price predictive signal while trading. When the trader creates an instantaneous market impact, it is shown that transaction costs of optimal adaptive strategies are substantially lower than the corresponding costs of the optimal static strategy. In the same spirit, in the case of transient impact it is shown that strategies that observe the signal a finite number of times can dramatically reduce the transaction costs and improve the performance of the optimal static strategy.

**A control problem for a speculative investor in a target zone model****Eyal Neuman, Alexander Schied**

We consider a stochastic control problem for a trader who wishes to maximize the expected local time through generating price impact. The local time can be regarded as a proxy for the inventory of a central bank whose aim is to maintain a target zone.

**An improved uniqueness result for a system of stochastic differential equations related to the stochastic wave equation****C. Mueller, E. Neuman, M. Salins, G. Truong**

We improve on the strong uniqueness results of [GLM+17], which deal with the following system of SDE.

dXtdYt=Ytdt=|Xt|αdBt

and X0=x0,Y0=y0. For (x0,y0)≠(0,0), we show that short-time uniqueness holds for α>−1/2.

**Hitting Probabilities of a Brownian flow with Radial Drift****Jong Jun Lee, Carl Mueller, Eyal Neuman**

We consider a stochastic flow ϕt(x,ω) in Rn with initial point ϕ0(x,ω)=x, driven by a single n-dimensional Brownian motion, and with an outward radial drift of magnitude F(∥ϕt(x)∥)∥ϕt(x)∥, with F nonnegative, bounded and Lipschitz. We consider initial points x lying in a set of positive distance from the origin. We show that there exist constants C∗,c∗>0 not depending on n, such that if F>C∗n then the image of the initial set under the flow has probability 0 of hitting the origin. If 0≤F≤c∗n3/4, and if the initial set has nonempty interior, then the image of the set has positive probability of hitting the origin.

**On the uniqueness of solutions of stochastic Volterra equations****Alexandre Pannier, Antoine Jacquier**

We prove strong existence and uniqueness, and Hölder regularity, of a large class of stochastic Volterra equations, with singular kernels and non-Lipschitz diffusion coefficient. Extending Yamada-Watanabe's theorem, our proof relies on an approximation of the process by a sequence of semimartingales with regularised kernels. We apply these results to the rough Heston model, with square-root diffusion coefficient, recently proposed in Mathematical Finance to model the volatility of asset prices

**A Quantum algorithm for linear PDEs arising in Finance**** Filipe Fontanela, Antoine Jacquier, Mugad Oumgari**

We propose a hybrid quantum-classical algorithm, originated from quantum chemistry, to price European and Asian options in the Black-Scholes model. Our approach is based on the equivalence between the pricing partial differential equation and the Schrodinger equation in imaginary time. We devise a strategy to build a shallow quantum circuit approximation to this equation, only requiring few qubits. This constitutes a promising candidate for the application of Quantum Computing techniques (with large number of qubits affected by noise) in Quantitative Finance.

**Dynamics of symmetric SSVI smiles and implied volatility bubbles****Mehdi El Amrani, Antoine Jacquier, Claude Martini**

We develop a dynamic version of the SSVI parameterisation for the total implied variance, ensuring that European vanilla option prices are martingales, hence preventing the occurrence of arbitrage, both static and dynamic. Insisting on the constraint that the total implied variance needs to be null at the maturity of the option, we show that no model--in our setting--allows for such behaviour. This naturally gives rise to the concept of implied volatility bubbles, whereby trading in an arbitrage-free way is only possible during part of the life of the contract, but not all the way until expiry.

**Anomalous diffusions in option prices: connecting trade duration and the volatility term structure****Antoine Jacquier, Lorenzo Torricelli**

Anomalous diffusions arise as scaling limits of continuous-time random walks (CTRWs) whose innovation times are distributed according to a power law. The impact of a non-exponential waiting time does not vanish with time and leads to different distribution spread rates compared to standard models. In financial modelling this has been used to accommodate for random trade duration in the tick-by-tick price process. We show here that anomalous diffusions are able to reproduce the market behaviour of the implied volatility more consistently than usual Lévy or stochastic volatility models. We focus on two distinct classes of underlying asset models, one with independent price innovations and waiting times, and one allowing dependence between these two components. These two models capture the well-known paradigm according to which shorter trade duration is associated with higher return impact of individual trades. We fully describe these processes in a semimartingale setting leading no-arbitrage pricing formulae, and study their statistical properties. We observe that skewness and kurtosis of the asset returns do not tend to zero as time goes by. We also characterize the large-maturity asymptotics of Call option prices, and find that the convergence rate is slower than in standard Lévy regimes, which in turn yields a declining implied volatility term structure and a slower decay of the skew.

**Deep PPDEs for rough local stochastic volatility****Antoine Jacquier, Mugad Oumgari**

We introduce the notion of rough local stochastic volatility models, extending the classical concept to the case where volatility is driven by some Volterra process. In this setting, we show that the pricing function is the solution to a path-dependent PDE, for which we develop a numerical scheme based on Deep Learning techniques. Numerical simulations suggest that the latter is extremely efficient, and provides a good alternative to classical Monte Carlo simulations.

**Stacked Monte Carlo for option pricing**** Antoine Jacquier, Emma R. Malone, Mugad Oumgari**

We introduce a stacking version of the Monte Carlo algorithm in the context of option pricing. Introduced recently for aeronautic computations, this simple technique, in the spirit of current machine learning ideas, learns control variates by approximating Monte Carlo draws with some specified function. We describe the method from first principles and suggest appropriate fits, and show its efficiency to evaluate European and Asian Call options in constant and stochastic volatility models.

**Small-time moderate deviations for the randomised Heston model****Antoine Jacquier, Fangwei Shi**

We extend previous large deviations results for the randomised Heston model to the case of moderate deviations. The proofs involve the Gärtner-Ellis theorem and sharp large deviations tools

**Perturbation analysis of sub/super hedging problems****Sergey Badikov, Mark H.A. Davis, Antoine Jacquier**

We investigate the links between various no-arbitrage conditions and the existence of pricing functionals in general markets, and prove the Fundamental Theorem of Asset Pricing therein. No-arbitrage conditions, either in this abstract setting or in the case of a market consisting of European Call options, give rise to duality properties of infinite-dimensional sub- and super-hedging problems. With a view towards applications, we show how duality is preserved when reducing these problems over finite-dimensional bases. We finally perform a rigorous perturbation analysis of those linear programming problems, and highlight numerically the influence of smile extrapolation on the bounds of exotic options.

**Statistical arbitrage of coherent risk measures****John Armstrong, Damiano Brigo**

We show that coherent risk measures are ineffective in curbing the behaviour of investors with limited liability if the market admits statistical arbitrage opportunities which we term ρ-arbitrage for a risk measure ρ. We show how to determine analytically whether such portfolios exist in complete markets and in the Markowitz model. We also consider realistic numerical examples of incomplete markets and determine whether expected shortfall constraints are ineffective in these markets. We find that the answer depends heavily upon the probability model selected by the risk manager but that it is certainly possible for expected shortfall constraints to be ineffective in realistic markets. Since value at risk constraints are weaker than expected shortfall constraints, our results can be applied to value at risk.

We investigate whether it is possible to formulate option pricing and hedging models without using probability. We present a model that is consistent with two notions of volatility: a historical volatility consistent with statistical analysis, and an implied volatility consistent with options priced with the model. The latter will be also the quadratic variation of the model, a pathwise property. This first result, originally presented in Brigo and Mercurio (1998, 2000), is then connected with the recent work of Armstrong et al (2018), where using rough paths theory it is shown that implied volatility is associated with a purely pathwise lift of the stock dynamics involving no probability and no semimartingale theory in particular, leading to option models without probability. Finally, an intermediate result by Bender et al. (2008) is recalled. Using semimartingale theory, Bender et al. showed that one could obtain option prices based only on the semimartingale quadratic variation of the model, a pathwise property, and highlighted the difference between historical and implied volatility. All three works confirm the idea that while historical volatility is a statistical quantity, implied volatility is a pathwise one. This leads to a 20 years mini-anniversary of pathwise pricing through 1998, 2008 and 2018, which is rather fitting for a talk presented at the conference for the 45 years of the Black, Scholes and Merton option pricing paradigm.

**On the consistency of jump-diffusion dynamics for FX rates under inversion****Federico Graceffa, Damiano Brigo, Andrea Pallavicini**

In this note we investigate the consistency under inversion of jump diffusion processes in the Foreign Exchange (FX) market. In other terms, if the EUR/USD FX rate follows a given type of dynamics, under which conditions will USD/EUR follow the same type of dynamics? In order to give a numerical description of this property, we first calibrate a Heston model and a SABR model to market data, plotting their smiles together with the smiles of the reciprocal processes. Secondly, we determine a suitable local volatility structure ensuring consistency. We subsequently introduce jumps and analyze both constant jump size (Poisson process) and random jump size (compound Poisson process). In the first scenario, we find that consistency is automatically satisfied, for the jump size of the inverted process is a constant as well. The second case is more delicate, since we need to make sure that the distribution of jumps in the domestic measure is the same as the distribution of jumps in the foreign measure. We determine a fairly general class of admissible densities for the jump size in the domestic measure satisfying the condition.

**Mechanics of good trade execution in the framework of linear temporary market impact****Claudio Bellani, Damiano Brigo**

We define the concept of good trade execution and we construct explicit adapted good trade execution strategies in the framework of linear temporary market impact. Good trade execution strategies are dynamic, in the sense that they react to the actual realisation of the traded asset price path over the trading period; this is paramount in volatile regimes, where price trajectories can considerably deviate from their expected value. Remarkably however, the implementation of our strategies does not require the full specification of an SDE evolution for the traded asset price, making them robust across different models. Moreover, rather than minimising the expected trading cost, good trade execution strategies minimise trading costs in a pathwise sense, a point of view not yet considered in the literature. The mathematical apparatus for such a pathwise minimisation hinges on certain random Young differential equations that correspond to the Euler-Lagrange equations of the classical Calculus of Variations. These Young differential equations characterise our good trade execution strategies in terms of an initial value problem that allows for easy implementations.

**Option pricing models without probability****John Armstrong, Claudio Bellani, Damiano Brigo, Thomas Cass**

We describe the pricing and hedging practices refraining from the use of probability. We encode volatility in an enhancement of the price trajectory and we give pathwise presentations of the fundamental equations of Mathematical Finance. In particular this allows us to assess model misspecification, generalising the so-called fundamental theorem of derivative trading (see Ellersgaard et al. 2017). Our pathwise integrals and equations exhibit the role of Greeks beyond the leading-order Delta, and makes explicit the role of Gamma sensitivities.

**Liquidity in Competitive Dealer Markets****Peter Bank, Ibrahim Ekren, Johannes Muhle-Karbe**

We study a continuous-time version of the intermediation model of Grossman and Miller (1988). To wit, we solve for the competitive equilibrium prices at which liquidity takers' demands are absorbed by dealers with quadratic inventory costs, who can in turn gradually transfer these positions to an end-user market. This endogenously leads to a model with transient price impact. Smooth, diffusive, and discrete trades all incur finite but nontrivial liquidity costs, and can arise naturally from the liquidity takers' optimization.

**Asset Pricing with Heterogenous Beliefs and Illiquidity****Johannes Muhle-Karbe, Marcel Nutz, Xiaowei Tan**

This paper studies the equilibrium price of an asset that is traded in continuous time between N agents who have heterogeneous beliefs about the state process underlying the asset's payoff. We propose a tractable model where agents maximize expected returns under quadratic costs on inventories and trading rates. The unique equilibrium price is characterized by a weakly coupled system of linear parabolic equations which shows that holding and liquidity costs play dual roles. We derive the leading-order asymptotics for small transaction and holding costs which give further insight into the equilibrium and the consequences of illiquidity.

**Asset Pricing with General Transaction Costs: Theory and Numerics****Lukas Gonon, Johannes Muhle-Karbe, Xiaofei Shi**

We study risk-sharing equilibria with general convex costs on the agents' trading rates. For an infinite-horizon model with linear state dynamics and exogenous volatilities, the equilibrium returns mean-revert around their frictionless counterparts -- the deviation has Ornstein-Uhlenbeck dynamics for quadratic costs whereas it follows a doubly-reflected Brownian motion if costs are proportional. More general models with arbitrary state dynamics and endogenous volatilities lead to multidimensional systems of nonlinear, fully-coupled forward-backward SDEs. These fall outside the scope of known wellposedness results, but can be solved numerically using the simulation-based deep-learning approach of \cite{han.al.17}. In a calibration to time series of returns, bid-ask spreads, and trading volume, transaction costs substantially affect equilibrium asset prices. In contrast, the effects of different cost specifications are rather similar, justifying the use of quadratic costs as a proxy for other less tractable specifications.

**Equilibrium Asset Pricing with Transaction Costs****Martin Herdegen, Johannes Muhle-Karbe, Dylan Possamaï**

We study risk-sharing economies where heterogenous agents trade subject to quadratic transaction costs. The corresponding equilibrium asset prices and trading strategies are characterised by a system of nonlinear, fully-coupled forward-backward stochastic differential equations. We show that a unique solution generally exists provided that the agents' preferences are sufficiently similar. In a benchmark specification with linear state dynamics, the illiquidity discounts and liquidity premia observed empirically correspond to a positive relationship between transaction costs and volatility.

**Manifold learning for accelerating coarse-grained optimization****Dmitry Pozharskiy, Noah J. Wichrowski, Andrew B. Duncan, Grigorios A. Pavliotis, Ioannis G. Kevrekidis**

Algorithms proposed for solving high-dimensional optimization problems with no derivative information frequently encounter the "curse of dimensionality," becoming ineffective as the dimension of the parameter space grows. One feature of a subclass of such problems that are effectively low-dimensional is that only a few parameters (or combinations thereof) are important for the optimization and must be explored in detail. Knowing these parameters/ combinations in advance would greatly simplify the problem and its solution. We propose the data-driven construction of an effective (coarse-grained, "trend") optimizer, based on data obtained from ensembles of brief simulation bursts with an "inner" optimization algorithm, that has the potential to accelerate the exploration of the parameter space. The trajectories of this "effective optimizer" quickly become attracted onto a slow manifold parameterized by the few relevant parameter combinations. We obtain the parameterization of this low-dimensional, effective optimization manifold on the fly using data mining/manifold learning techniques on the results of simulation (inner optimizer iteration) burst ensembles and exploit it locally to "jump" forward along this manifold. As a result, we can bias the exploration of the parameter space towards the few, important directions and, through this "wrapper algorithm," speed up the convergence of traditional optimization algorithms.

**A proof of the mean-field limit for λ-convex potentials using Γ-convergence****J. A. Carrillo, M. G. Delgadino, G. A. Pavliotis**

In this work we give a proof of the mean-field limit for λ-convex potentials using a purely variational viewpoint. Our approach is based on the observation that all evolution equations that we study can be written as gradient flows of functionals at different levels: in the set of probability measures, in the set of symmetric probability measures on N variables, and in the set of probability measures on probability measures. This basic fact allows us to rely on Γ-convergence tools for gradient flows to complete the proof by identifying the limits of the different terms in the Evolutionary Variational Inequalities (EVIs) associated to each gradient flow. The λ-convexity of the confining and interaction potentials is crucial for the unique identification of the limits and for deriving the EVIs at each description level of the interacting particle system.

**The sharp, the flat and the shallow: Can weakly interacting agents learn to escape bad minima?****Nikolas Kantas, Panos Parpas, Grigorios A. Pavliotis**

An open problem in machine learning is whether flat minima generalize better and how to compute such minima efficiently. This is a very challenging problem. As a first step towards understanding this question we formalize it as an optimization problem with weakly interacting agents. We review appropriate background material from the theory of stochastic processes and provide insights that are relevant to practitioners. We propose an algorithmic framework for an extended stochastic gradient Langevin dynamics and illustrate its potential. The paper is written as a tutorial, and presents an alternative use of multi-agent learning. Our primary focus is on the design of algorithms for machine learning applications; however the underlying mathematical framework is suitable for the understanding of large scale systems of agent based models that are popular in the social sciences, economics and finance.

**Mean field limits for interacting diffusions with colored noise: phase transitions and spectral numerical methods****S. N. Gomes, G. A. Pavliotis, U. Vaes**

In this paper we consider systems of weakly interacting particles driven by colored noise in a bistable potential, and we study the effect of the correlation time of the noise on the bifurcation diagram for the equilibrium states. We accomplish this by solving the corresponding McKean-Vlasov equation using a Hermite spectral method, and we verify our findings using Monte Carlo simulations of the particle system. We consider both Gaussian and non-Gaussian noise processes, and for each model of the noise we also study the behavior of the system in the small correlation time regime using perturbation theory. The spectral method that we develop in this paper can be used for solving linear and nonlinear, local and nonlocal (mean-field) Fokker-Planck equations, without requiring that they have a gradient structure.

**Long-time behaviour and phase transitions for the McKean-Vlasov equation on the torus****J. A. Carrillo, R. S. Gvalani, G. A. Pavliotis, A. Schlichting**

We study the McKean-Vlasov equation ∂tϱ=β−1Δϱ+κ∇⋅(ϱ∇(W⋆ϱ)), with periodic boundary conditions on the torus. We first study the global asymptotic stability of the homogeneous steady state. We then focus our attention on the stationary system, and prove the existence of nontrivial solutions branching from the homogeneous steady state, through possibly infinitely many bifurcations, under appropriate assumptions on the interaction potential. We also provide sufficient conditions for the existence of continuous and discontinuous phase transitions. Finally, we showcase these results by applying them to several examples of interaction potentials such as the noisy Kuramoto model for synchronisation, the Keller--Segel model for bacterial chemotaxis, and the noisy Hegselmann--Krausse model for opinion dynamics.

**Accelerated convergence to equilibrium and reduced asymptotic variance for Langevin dynamics using Stratonovich perturbations****Assyr Abdulle, Grigorios A. Pavliotis, Gilles Vilmart**

In this paper we propose a new approach for sampling from probability measures in, possibly, high dimensional spaces. By perturbing the standard overdamped Langevin dynamics by a suitable Stratonovich perturbation that preserves the invariant measure of the original system, we show that accelerated convergence to equilibrium and reduced asymptotic variance can be achieved, leading, thus, to a computationally advantageous sampling algorithm. The new perturbed Langevin dynamics is reversible with respect to the target probability measure and, consequently, does not suffer from the drawbacks of the nonreversible Langevin samplers that were introduced in~[C.-R. Hwang, S.-Y. Hwang-Ma, and S.-J. Sheu, Ann. Appl. Probab. 1993] and studied in, e.g. [T. Lelievre, F. Nier, and G. A. Pavliotis J. Stat. Phys., 2013] and [A. B. Duncan, T. Lelièvre, and G. A. Pavliotis J. Stat. Phys., 2016], while retaining all of their advantages in terms of accelerated convergence and reduced asymptotic variance. In particular, the reversibility of the dynamics ensures that there is no oscillatory transient behaviour. The improved performance of the proposed methodology, in comparison to the standard overdamped Langevin dynamics and its nonreversible perturbation, is illustrated on an example of sampling from a two-dimensional warped Gaussian target distribution

**Dynamics of the Desai-Zwanzig model in multi-well and random energy landscapes****Susana N. Gomes, Serafim Kalliadasis, Grigorios A. Pavliotis, Petr Yatsyshin**

We analyze a variant of the Desai-Zwanzig model [J. Stat. Phys. {\bf 19}1-24 (1978)]. In particular, we study stationary states of the mean field limit for a system of weakly interacting diffusions moving in a multi-well potential energy landscape, coupled via a Curie-Weiss type (quadratic) interaction potential. The location and depth of the local minima of the potential are either deterministic or random. We characterize the structure and nature of bifurcations and phase transitions for this system, by means of extensive numerical simulations and of analytical calculations for an explicitly solvable model. Our numerical experiments are based on Monte Carlo simulations, the numerical solution of the time-dependent nonlinear Fokker-Planck (McKean-Vlasov equation), the minimization of the free energy functional and a continuation algorithm for the stationary solutions.

**Constructing sampling schemes via coupling: Markov semigroups and optimal transport****N. Nuesken, G. A. Pavliotis**

In this paper we develop a general framework for constructing and analysing coupled Markov chain Monte Carlo samplers, allowing for both (possibly degenerate) diffusion and piecewise deterministic Markov processes. For many performance criteria of interest, including the asymptotic variance, the task of finding efficient couplings can be phrased in terms of problems related to optimal transport theory. We investigate general structural properties, proving a singularity theorem that has both geometric and probabilistic interpretations. Moreover, we show that those problems can often be solved approximately and support our findings with numerical experiments. For the particular objective of estimating the variance of a Bayesian posterior, our analysis suggests using novel techniques in the spirit of antithetic variates. Addressing the convergence to equilibrium of coupled processes we furthermore derive a modified Poincaré inequality.

**Instability, rupture and fluctuations in thin liquid films: Theory and computations****Miguel A. Durán-Olivencia, Rishabh S. Gvalani, Serafim Kalliadasis, Grigorios A. Pavliotis**

Thin liquid films are ubiquitous in natural phenomena and technological applications. They have been extensively studied via deterministic hydrodynamic equations, but thermal fluctuations often play a crucial role that needs to be understood. An example of this is dewetting, which involves the rupture of a thin liquid film and the formation of droplets. Such a process is thermally activated and requires fluctuations to be taken into account self-consistently. In this work we present an analytical and numerical study of a stochastic thin-film equation derived from first principles. Following a brief review of the derivation, we scrutinise the behaviour of the equation in the limit of perfectly correlated noise along the wall-normal direction. The stochastic thin-film equation is also simulated by adopting a numerical scheme based on a spectral collocation method. The scheme allows us to explore the fluctuating dynamics of the thin film and the behaviour of its free energy in the vicinity of rupture. Finally, we also study the effect of the noise intensity on the rupture time, which is in agreement with previous works.

**A priori bounds for the $\Phi^4$ equation in the full sub-critical regime ****Ajay Chandra, Augustin Moinat, Hendrik Weber**

We derive a priori bounds for the Φ4 equation in the full sub-critical regime using Hairer's theory of regularity structures. The equation is formally given by (∂t−Δ)ϕ=−ϕ3+∞ϕ+ξ,(⋆) where the term +∞ϕ represents infinite terms that have to be removed in a renormalisation procedure. (...)

Log-Hessian Formula And The Talagrand Conjecture**N Gozlan, Xue-Mei Li, M Madiman, C Roberto, P M Samson**

In 1989, Talagrand proposed a conjecture regarding the regularization effect on integrable functions of a natural Markov semigroup on the Boolean hypercube. While this conjecture remains unresolved, the analogous conjecture for the Ornstein-Uhlenbeck semigroup was recently resolved by Eldan-Lee and Lehec, by combining an inequality for the log-Hessian of this semigroup with a new deviation inequality for log-semiconvex functions under Gaussian measure. Our first goal is to explore the validity of both these ingredients for some diffusion semigroups in Rn as well as for the M/M/OE queue on the non-negative integers. Our second goal is to prove an analogue of Talagrand’s conjecture for these settings, even in those cases where these ingredients are not valid.

Homogenization with fractional random fields**Johann Gehringer, Xue-Mei Li**

We consider a system of differential equations in a fast long range dependent random environment and prove a homogenization theorem involving multiple scaling constants. The effective dynamics solves a rough differential equation, which is ‘equivalent’ to a stochasticequation driven by mixed Itô integrals and Young integrals with respect to Wiener processes and Hermite processes. Lacking other tools we use the rough path theory for proving the convergence, our main technical endeavour is on obtaining an enhanced scaling limit theorem for path integrals (Functional CLT and non-CLT’s) in a strong topology, the rough path topology, which is given by a Hölder distance for stochastic processes and their lifts. In

dimension one we also include the negatively correlated case, for the second order / kinetic fractional BM model we also bound the error.

### 2018

**Projections of SDEs onto Submanifolds****John Armstrong, Damiano Brigo, Emilio Rossi Ferrucci**

We describe the pricing and hedging practices refraining from the use of probability. We encode volatility in an enhancement of the price trajectory and we give pathwise presentations of the fundamental equations of Mathematical Finance. In particular this allows us to assess model misspecification, generalising the so-called fundamental theorem of derivative trading (see Ellersgaard et al. 2017). Our pathwise integrals and equations exhibit the role of Greeks beyond the leading-order Delta, and makes explicit the role of Gamma sensitivities.

**Risk-neutral valuation under differential funding costs, defaults and collateralizationDamiano Brigo, Cristin Buescu, Marco Francischello, Andrea Pallavicini, Marek Rutkowski**

We develop a unified valuation theory that incorporates credit risk (defaults), collateralization and funding costs, by expanding the replication approach to a generality that has not yet been studied previously and reaching valuation when replication is not assumed. This unifying theoretical framework clarifies the relationship between the two valuation approaches: the adjusted cash flows approach pioneered for example by Brigo, Pallavicini and co-authors ([12, 13, 34]) and the classic replication approach illustrated for example by Bielecki and Rutkowski and co-authors ([3, 8]). In particular, results of this work cover most previous papers where the authors studied specific replication models.

**Numerically Modelling Stochastic Lie Transport in Fluid Dynamics**

**Colin J. Cotter, Dan Crisan, Darryl D. Holm, Wei Pan, Igor Shevchenko**

We present a numerical investigation of stochastic transport in ideal fluids. According to Holm (Proc Roy Soc, 2015) and Cotter et al. (2017), the principles of transformation theory and multi-time homogenisation, respectively, imply a physically meaningful, data-driven approach for decomposing the fluid transport velocity into its drift and stochastic parts, for a certain class of fluid flows. In the current paper, we develop new methodology to implement this velocity decomposition and then numerically integrate the resulting stochastic partial differential equation using a finite element discretisation for incompressible 2D Euler fluid flows. The new methodology tested here is found to be suitable for coarse graining in this case. Specifically, we perform uncertainty quantification tests of the velocity decomposition of Cotter et al. (2017), by comparing ensembles of coarse-grid realisations of solutions of the resulting stochastic partial differential equation with the "true solutions" of the deterministic fluid partial differential equation, computed on a refined grid. The time discretization used for approximating the solution of the stochastic partial differential equation is shown to be consistent. We include comprehensive numerical tests that confirm the non-Gaussianity of the stream function, velocity and vorticity fields in the case of incompressible 2D Euler fluid flows.

**Modelling uncertainty using circulation-preserving stochastic transport noise in a 2-layer quasi-geostrophic model**

**Colin J. Cotter, Dan Crisan, Darryl D. Holm, Wei Pan, Igor Shevchenko**

The stochastic variational approach for geophysical fluid dynamics was introduced by Holm (Proc Roy Soc A, 2015) as a framework for deriving stochastic parameterisations for unresolved scales. The key feature of transport noise is that it respects the Kelvin circulation theorem. This paper applies the variational stochastic parameterisation in a two-layer quasi-geostrophic model for a β-plane channel flow configuration. The parameterisation is tested by comparing it with a deterministic high resolution eddy-resolving solution that has reached statistical equilibrium. We describe a stochastic time-stepping scheme for the two-layer model and discuss its consistency in time. Then we describe a procedure for estimating the stochastic forcing to approximate unresolved components using data from the high resolution deterministic simulation. We compare an ensemble of stochastic solutions at lower resolution with the numerical solution of the deterministic model. These computations quantify the uncertainty of the coarse grid computation relative to the fine grid computation. The results show that the proposed parameterisation is efficient and effective for both homogeneous and heterogeneous flows, and they lay a solid foundation for data assimilation.

**A Stratonovich-Skorohod integral formula for Volterra Gaussian rough paths**

**Thomas Cass, Nengli Lim**

Given a solution Y to a rough differential equation (RDE), a recent result [8] extends the classical Itö-Stratonovich formula and provides a closed-form expression for ∫Y∘dX−∫YdX, i.e. the difference between the rough and Skorohod integrals of Y with respect to X, where X is a Gaussian process with finite p-variation less than 3. In this paper, we extend this result to Gaussian processes with finite p-variation such that 3≤p<4. The constraint this time is that we restrict ourselves to Volterra Gaussian processes with kernels satisfying a natural condition, which however still allows the result to encompass many standard examples, including fractional Brownian motion with H>14. Analogously to [8], we first show that the Riemann-sum approximants of the Skorohod integral converge in L2(Ω) by adopting a suitable characterization of the Cameron-Martin norm, before appending the approximants with higher-level compensation terms without altering the limit. Lastly, the formula is obtained after a re-balancing of terms, and we also show how to recover the standard Itö formulas in the case where the vector fields of the RDE governing Y are commutative.

**Long-time behaviour of degenerate diffusions: UFG-type SDEs and time-inhomogeneous hypoelliptic processes**

**Thomas Cass, Dan Crisan, Paul Dobson, Michela Ottobre**

We study the long time behaviour of a large class of diffusion processes on R^N, generated by second order differential operators of (possibly) degenerate type. The operators that we consider need not satisfy the Hormander condition. Instead, they satisfy the so-called UFG condition, introduced by Herman, Lobry and Sussman in the context of geometric control theory and later by Kusuoka and Stroock, this time with probabilistic motivations. In this paper we will demonstrate the importance of UFG diffusions in several respects: roughly speaking i) we show that UFG processes constitute a family of SDEs which exhibit multiple invariant measures and for which one is able to describe a systematic procedure to determine the basin of attraction of each invariant measure (equilibrium state). ii)We show that our results and techniques, which we devised for UFG processes, can be applied to the study of the long-time behaviour of non-autonomous hypoelliptic SDEs. iii) We prove that there exists a change of coordinates such that every UFG diffusion can be, at least locally, represented as a system consisting of an SDE coupled with an ODE, where the ODE evolves independently of the SDE part of the dynamics. iv) As a result, UFG diffusions are inherently less smooth" than hypoelliptic SDEs; more precisely, we prove that UFG processes do not admit a density with respect to Lebesgue measure on the entire space, but only on suitable time-evolving submanifolds, which we describe.

**Large-scale limit of interface fluctuation models**

**Martin Hairer, Weijun Xu**

We extend the weak universality of KPZ in [Hairer-Quastel] to weakly asymmetric interface models with general growth mechanisms beyond polynomials. A key new ingredient is a pointwise bound on correlations of trigonometric functions of Gaussians in terms of their polynomial counterparts. This enables us to reduce the problem of a general nonlinearity with sufficient regularity to that of a polynomial.

**Volatility options in rough volatility models**

**Blanka Horvath, Antoine Jacquier, Peter Tankov**

We discuss the pricing and hedging of volatility options in some rough volatility models. First, we develop efficient Monte Carlo methods and asymptotic approximations for computing option prices and hedge ratios in models where log-volatility follows a Gaussian Volterra process. While providing a good fit for European options, these models are unable to reproduce the VIX option smile observed in the market, and are thus not suitable for VIX products. To accommodate these, we introduce the class of modulated Volterra processes, and show that they successfully capture the VIX smile.

**Dirichlet Forms and Finite Element Methods for the SABR Model**

**Blanka Horvath, Oleg Reichmann**

We propose a deterministic numerical method for pricing vanilla options under the SABR stochastic volatility model, based on a finite element discretization of the Kolmogorov pricing equations via non-symmetric Dirichlet forms. Our pricing method is valid under mild assumptions on parameter configurations of the process both in moderate interest rate environments and in near-zero interest rate regimes such as the currently prevalent ones. The parabolic Kolmogorov pricing equations for the SABR model are degenerate at the origin, yielding non-standard partial differential equations, for which conventional pricing methods ---designed for non-degenerate parabolic equations--- potentially break down. We derive here the appropriate analytic setup to handle the degeneracy of the model at the origin. That is, we construct an evolution triple of suitably chosen Sobolev spaces with singular weights, consisting of the domain of the SABR-Dirichlet form, its dual space, and the pivotal Hilbert space. In particular, we show well-posedness of the variational formulation of the SABR-pricing equations for vanilla and barrier options on this triple. Furthermore, we present a finite element discretization scheme based on a (weighted) multiresolution wavelet approximation in space and a θ-scheme in time and provide an error analysis for this discretization.

**Perturbation analysis of sub/super hedging problems**

**Sergey Badikov, Mark H. A. Davis, Antoine Jacquier**

We investigate the links between various no-arbitrage conditions and the existence of pricing functionals in general markets, and prove the Fundamental Theorem of Asset Pricing therein. No-arbitrage conditions, either in this abstract setting or in the case of a market consisting of European Call options, give rise to duality properties of infinite-dimensional sub- and super-hedging problems. With a view towards applications, we show how duality is preserved when reducing these problems over finite-dimensional bases. We finally perform a rigorous perturbation analysis of those linear programming problems, and highlight, as a numerical example, the influence of smile extrapolation on the bounds of exotic options.

**Pathwise moderate deviations for option pricing**

**Antoine Jacquier, Konstantinos Spiliopoulos**

We provide a unifying treatment of pathwise moderate deviations for models commonly used in financial applications, and for related integrated functionals. Suitable scaling allows us to transfer these results into small-time, large-time and tail asymptotics for diffusions, as well as for option prices and realised variances. In passing, we highlight some intuitive relationships between moderate deviations rate functions and their large deviations counterparts; these turn out to be useful for numerical purposes, as large deviations rate functions are often difficult to compute.

**Protecting Target Zone Currency Markets from Speculative Investors**

**Eyal Neuman, Alexander Schied**

We consider a stochastic game between a trader and the central bank on target zone markets. In this type of markets the price process is modeled as a diffusion which is reflected at one or more barriers. Such models arise when a currency exchange rate is kept above a certain threshold due to central bank intervention. We consider a trader who wishes to liquidate a large amount of currency, where for whom prices are optimal at the barrier. The central bank, who wishes to keep the currency exchange rate above this barrier, therefore needs to buy its own currency. The permanent price impact, which is created by the transactions of both sides, turns the optimal trading problems of the trader and the central bank into coupled singular control problems, where the common singularity arises from a local time along a random curve. We first solve the central bank's control problem by means of the Skorokhod map and then derive the trader's optimal strategy by solving a sequence of approximated control problems, thus establishing a Stackelberg equilibrium in our model.

**Hitting Probabilities of a Brownian flow with Radial Drift**

**Jong Jun Lee, Carl Mueller, Eyal Neuman**

We consider a stochastic flow φt(x,ω) in Rn with initial point φ0(x,ω)=x, driven by a single n-dimensional Brownian motion, and with an outward radial drift of magnitude F(∥φt(x)∥)∥φt(x)∥, with F nonnegative, bounded and Lipschitz. We consider initial points x lying in a ball of positive distance from the origin. We show that there exist constants C∗,c∗>0 not depending on n, such that if F>C∗n then the image of the initial ball under the flow has probability 0 of hitting the origin. If 0≤F<c∗n/logn, then the image of the ball has positive probability of hitting the origin.

**Recent advances in the evolution of interfaces: thermodynamics, upscaling, and universality**

**M. Schmuck, G. A. Pavliotis, S. Kalliadasis**

We consider the evolution of interfaces in binary mixtures permeating strongly heterogeneous systems such as porous media. To this end, we first review available thermodynamic formulations for binary mixtures based on \emph{general reversible-irreversible couplings} and the associated mathematical attempts to formulate a \emph{non-equilibrium variational principle} in which these non-equilibrium couplings can be identified as minimizers. Based on this, we investigate two microscopic binary mixture formulations fully resolving heterogeneous/perforated domains: (a) a flux-driven immiscible fluid formulation without fluid flow; (b) a momentum-driven formulation for quasi-static and incompressible velocity fields. In both cases we state two novel, reliably upscaled equations for binary mixtures/multiphase fluids in strongly heterogeneous systems by systematically taking thermodynamic features such as free energies into account as well as the system's heterogeneity defined on the microscale such as geometry and materials (e.g. wetting properties). In the context of (a), we unravel a \emph{universality} with respect to the coarsening rate due to its independence of the system's heterogeneity, i.e. the well-known O(t1/3)-behaviour for homogeneous systems holds also for perforated domains. Finally, the versatility of phase field equations and their \emph{thermodynamic foundation} relying on free energies, make the collected recent developments here highly promising for scientific, engineering and industrial applications for which we provide an example for lithium batteries.

**Stochastic Modelling of Urban Structure**

**L. Ellam, M. Girolami, G. A. Pavliotis, A. Wilson**

The building of mathematical and computer models of cities has a long history. The core elements are models of flows (spatial interaction) and the dynamics of structural evolution. In this article, we develop a stochastic model of urban structure to formally account for uncertainty arising from less predictable events. Standard practice has been to calibrate the spatial interaction models independently and to explore the dynamics through simulation. We present two significant results that will be transformative for both elements. First, we represent the structural variables through a single potential function and develop stochastic differential equations (SDEs) to model the evolution. Secondly, we show that the parameters of the spatial interaction model can be estimated from the structure alone, independently of flow data, using the Bayesian inferential framework. The posterior distribution is doubly intractable and poses significant computational challenges that we overcome using Markov chain Monte Carlo (MCMC) methods. We demonstrate our methodology with a case study on the London retail system

**Mean field limits for non-Markovian interacting particles: convergence to equilibrium, GENERIC formalism, asymptotic limits and phase transitions**

**M. H. Duong, G. A. Pavliotis**

In this paper, we study the mean field limit of interacting particles with memory that are governed by a system of interacting non-Markovian Langevin equations. Under the assumption of quasi-Markovianity (i.e. that the memory in the system can be described using a finite number of auxiliary processes), we pass to the mean field limit to obtain the corresponding McKean-Vlasov equation in an extended phase space. We obtain the fundamental solution (Green's function) for this equation, for the case of a quadratic confining potential and a quadratic (Curie-Weiss) interaction. Furthermore, for nonconvex confining potentials we characterize the stationary state(s) of the McKean-Vlasov equation, and we show that the bifurcation diagram of the stationary problem is independent of the memory in the system. In addition, we show that the McKean-Vlasov equation for the non-Markovian dynamics can be written in the GENERIC formalism and we study convergence to equilibrium and the Markovian asymptotic limit.

**Optimal control of thin liquid films and transverse mode effects**

**Ruben J. Tomlin, Susana N. Gomes, Grigorios A. Pavliotis, Demetrios T. Papageorgiou**

We consider the control of a three-dimensional thin liquid film on a flat substrate, inclined at a non-zero angle to the horizontal. Controls are applied via same-fluid blowing and suction through the substrate surface. We consider both overlying and hanging films, where the liquid lies above or below the substrate, respectively. We study the weakly nonlinear evolution of the system, which is governed by a forced Kuramoto--Sivashinsky equation in two space dimensions. The uncontrolled problem exhibits three ranges of dynamics depending on the incline of the substrate: stable flat film solution, bounded chaotic dynamics, or unbounded exponential growth of unstable transverse modes. We proceed with the assumption that we may actuate at every point on the substrate. The main focus is the optimal control problem, which we first study in the special case that the forcing may only vary in the spanwise direction. The structure of the Kuramoto--Sivashinsky equation allows the explicit construction of optimal controls in this case using the classical theory of linear quadratic regulators. Such controls are employed to prevent the exponential growth of transverse waves in the case of a hanging film, revealing complex dynamics for the streamwise and mixed modes. We then consider the optimal control problem in full generality, and prove the existence of an optimal control. For numerical simulations, we employ an iterative gradient descent algorithm. In the final section, we consider the effects of transverse mode forcing on the chaotic dynamics present in the streamwise and mixed modes for the case of a vertical film flow. Coupling through nonlinearity allows us to reduce the average energy in solutions without directly forcing the linearly unstable dominant modes.

**Long-time behaviour and phase transitions for the McKean--Vlasov equation on the torus**

**J. A. Carrillo, R. S. Gvalani, G. A. Pavliotis, A. Schlichting**

We study the McKean-Vlasov equation, ∂tϱ=β−1Δϱ+κ∇⋅(ϱ∇(W⋆ϱ)), with periodic boundary conditions on the torus. We first study the global asymptotic stability of the homogeneous steady state. We then focus our attention on the stationary system, and prove the existence of nontrivial solutions branching from the homogeneous steady state, through possibly infinitely many bifurcations, under appropriate assumptions on the interaction potential. We also provide sufficient conditions for the existence of continuous and discontinuous phase transitions. Finally, we showcase these results by applying them to several examples of interaction potentials such as the noisy Kuramoto model for synchronisation, the Keller--Segel model for bacterial chemotaxis, and the noisy Hegselmann--Krausse model for opinion dynamics.

**Constructing sampling schemes via coupling: Markov semigroups and optimal transport**

**N. Nuesken, G. A. Pavliotis**

In this paper we develop a general framework for constructing and analysing coupled Markov chain Monte Carlo samplers, allowing for both (possibly degenerate) diffusion and piecewise deterministic Markov processes. For many performance criteria of interest, including the asymptotic variance, the task of finding efficient couplings can be phrased in terms of problems related to optimal transport theory. We investigate general structural properties, proving a singularity theorem that has both geometric and probabilistic interpretations. Moreover, we show that those problems can often be solved approximately and support our findings with numerical experiments. For the particular objective of estimating the variance of a Bayesian posterior, our analysis suggests using novel techniques in the spirit of antithetic variates. Addressing the convergence to equilibrium of coupled processes we furthermore derive a modified Poincaré inequality.

**Early-warning signals for bifurcations in random dynamical systems with bounded noise**

**Christian Kuehn, Giuseppe Malavolta, Martin Rasmussen**

We consider discrete-time one-dimensional random dynamical systems with bounded noise, which generate an associated set-valued dynamical system. We provide necessary and sufficient conditions for a discontinuous bifurcation of a minimal invariant set of the set-valued dynamical system in terms of the derivatives of the so-called extremal maps. We propose an algorithm for reconstructing the derivatives of the extremal maps from a time series that is generated by iterations of the original random dynamical system. We demonstrate that the derivative reconstructed for different parameters can be used as an early-warning signal to detect an upcoming bifurcation, and apply the algorithm to the bifurcation analysis of the stochastic return map of the Koper model, which is a three-dimensional multiple time scale ordinary differential equation used as prototypical model for the formation of mixed-mode oscillation patterns. We apply our algorithm to data generated by this map to detect an upcoming transition.

**Conditioned Lyapunov exponents for random dynamical systems**

**Maximilian Engel, Jeroen S. W. Lamb, Martin Rasmussen**

We introduce the notion of Lyapunov exponents for random dynamical systems, conditioned to trajectories that stay within a bounded domain for asymptotically long times. This is motivated by the desire to characterize local dynamical properties in the presence of unbounded noise (when almost all trajectories are unbounded). We illustrate its use in the analysis of local bifurcations in this context. The theory of conditioned Lyapunov exponents of stochastic differential equations builds on the stochastic analysis of quasi-stationary distributions for killed processes and associated quasi-ergodic distributions. We show that conditioned Lyapunov exponents describe the local stability behaviour of trajectories that remain within a bounded domain and - in particular - that negative conditioned Lyapunov exponents imply local synchronisation. Furthermore, a conditioned dichotomy spectrum is introduced and its main characteristics are established.

**Quasi-shuffle algebras and renormalisation of rough differential equations**

**Yvain Bruned, Charles Curry, Kurusch Ebrahimi-Fard**

The objective of this work is to compare several approaches to the process of renormalisation in the context of rough differential equations using the substitution bialgebra on rooted trees known from backward error analysis of B-series. For this purpose, we present a so-called arborification of the Hoffman--Ihara theory of quasi-shuffle algebra automorphisms. The latter are induced by formal power series, which can be seen to be special cases of the cointeraction of two Hopf algebra structures on rooted forests. In particular, the arborification of Hoffman's exponential map, which defines a Hopf algebra isomorphism between the shuffle and quasi-shuffle Hopf algebra, leads to a canonical renormalisation that coincides with Marcus' canonical extension for semimartingale driving signals. This is contrasted with the canonical geometric rough path of Hairer and Kelly by means of a recursive formula defined in terms of the coaction of the substitution bialgebra.

**Large permutation invariant random matrices are asymptotically free over the diagonal**

**Benson Au, Guillaume Cébron, Antoine Dahlqvist, Franck Gabriel, Camille Male**

We prove that independent families of permutation invariant random matrices are asymptotically free over the diagonal, both in probability and in expectation, under a uniform boundedness assumption on the operator norm. We can relax the operator norm assumption to an estimate on sums associated to graphs of matrices, further extending the range of applications (for example, to Wigner matrices with exploding moments and so the sparse regime of the Erdős-Rényi model). The result still holds even if the matrices are multiplied entrywise by bounded random variables (for example, as in the case of matrices with a variance profile and percolation models).

**Neural Tangent Kernel: Convergence and Generalization in Neural Networks**

**Arthur Jacot, Franck Gabriel, Clément Hongler**

At initialization, artificial neural networks (ANNs) are equivalent to Gaussian processes in the infinite-width limit, thus connecting them to kernel methods. We prove that the evolution of an ANN during training can also be described by a kernel: during gradient descent on the parameters of an ANN, the network function fθ (which maps input vectors to output vectors) follows the kernel gradient of the functional cost (which is convex, in contrast to the parameter cost) w.r.t. a new kernel: the Neural Tangent Kernel (NTK). This kernel is central to describe the generalization features of ANNs. While the NTK is random at initialization and varies during training, in the infinite-width limit it converges to an explicit limiting kernel and it stays constant during training. This makes it possible to study the training of ANNs in function space instead of parameter space. Convergence of the training can then be related to the positive-definiteness of the limiting NTK. We prove the positive-definiteness of the limiting NTK when the data is supported on the sphere and the non-linearity is non-polynomial. We then focus on the setting of least-squares regression and show that in the infinite-width limit, the network function fθfollows a linear differential equation during training. The convergence is fastest along the largest kernel principal components of the input data with respect to the NTK, hence suggesting a theoretical motivation for early stopping. Finally we study the NTK numerically, observe its behavior for wide networks, and compare it to the infinite-width limit.

**On the support of solutions of stochastic differential equations with path-dependent coefficients**

**Rama Cont, Alexander Kalinin**

Given a stochastic differential equation with path-dependent coefficients driven by a multidimensional Wiener process, we show that the support of the law of the solution is given by the image of the Cameron-Martin space under the flow of the solutions of a system of path-dependent (ordinary) differential equations. Our result extends the Stroock-Varadhan support theorem for diffusion processes to the case of SDEs with path-dependent coefficients. The proof is based on the Functional Ito calculus and interpolation estimates for stochastic processes in Holder norm.

**Asymptotics of the density of parabolic Anderson random fields**

**Yaozhong Hu, Khoa Lê**

We investigate the sharp density ρ(t,x;y) of the solution u(t,x) to stochastic partial differential equation ∂∂tu(t,x)=12Δu(t,x)+u⋄W˙(t,x), where W˙ is a general Gaussian noise and ⋄ denotes the Wick product. We mainly concern with the asymptotic behavior of ρ(t,x;y) when y→∞ or when t→0+. Both upper and lower bounds are obtained and these two bounds match each other modulo some multiplicative constants. If the initial datum is positive, then ρ(t,x;y) is supported on the positive half line y∈[0,∞) and in this case we show that ρ(t,x;0+)=0 and obtain an upper bound for ρ(t,x;y) when y→0+

**Perturbation of Conservation Laws and Averaging on Manifolds**

**Xue-Mei Li**

We prove a stochastic averaging theorem for stochastic differential equa- tions in which the slow and the fast variables interact. The approximate Markov fast motion is a family of Markov process with generator Lx for which we obtain a locally uniform law of large numbers and obtain the continuous dependence of their invariant measures on the parameter x. These results are obtained under the as- sumption that Lx satisfies Ho ̈rmander’s bracket conditions, or more generally Lx is a family of Fredholm operators with sub-elliptic estimates. On the other hand a con- servation law of a dynamical system can be used as a tool for separating the scales in singular perturbation problems. We also study a number of motivating examples from mathematical physics and from geometry where we use non-linear conserva- tion laws to deduce slow-fast systems of stochastic differential equations.

**Geometry of Gaussian free field sign clusters and random interlacements****Alexander Drewitz, Alexis Prévost, Pierre-François Rodriguez**

For a large class of amenable transient weighted graphs G, we prove that the sign clusters of the Gaussian free field on G fall into a regime of strong supercriticality, in which two infinite sign clusters dominate (one for each sign), and finite sign clusters are necessarily tiny, with overwhelming probability. Examples of graphs belonging to this class include regular lattices like Zd, for d⩾3, but also more intricate geometries, such as Cayley graphs of suitably growing (finitely generated) non-Abelian groups, and cases in which random walks exhibit anomalous diffusive behavior, for instance various fractal graphs. As a consequence, we also show that the vacant set of random interlacements on these objects, introduced by Sznitman in arXiv:0704.2560, and which is intimately linked to the free field, contains an infinite connected component at small intensities. In particular, this result settles an open problem from arXiv:1010.1490.

### 2017

17-01

Archil Gulisashvili, Blanka Horvath, Antoine Jacquier

**Mass At Zero In The Uncorrelated Sabr Model And Implied Volatility Asymptotics.**

We study the mass at the origin in the uncorrelated SABR stochastic volatility model, and derive several tractable expressions, in particular when time becomes small or large.

As an application{in fact the original motivation for this paper{we derive small-strike expansions for the implied volatility when the maturity becomes short or large. These formulae, by denition arbitrage free, allow us to quantify the impact of the mass at zero on existing implied volatility approximations, and in particular how correct/erroneous these approximations become.

17-02

Hamza Guennoun, Antoine Jacquier, Patrick Roome, And Fangwei Shi

**Asymptotic Behaviour Of The Fractional Heston Model.**

We consider the fractional Heston model originally proposed by Comte, Coutin and Renault [12]. Inspired by recent ground-breaking work on rough volatility [2, 6, 24, 26] which showed that models with volatility driven by fractional Brownian motion with short memory allows for better calibration of the volatility surface and more robust estimation of time series of historical volatility, we provide a characterisation of the short- and long-maturity asymptotics of the implied volatility smile. Our analysis reveals that the short-memory property precisely provides a jump-type behaviour of the smile for short maturities, thereby _xing the well-known standard inability of classical stochastic volatility models to _t the short-end of the volatility smile.

17-03

Blanka Horvath, Antoine Jacquier, Chlo_E Lacombe**Asymptotic Behaviour Of Randomised Fractional Volatility Models.**

We study the asymptotic behaviour of a class of small-noise di_usions driven by fractional Brownian motion, with random starting points. Di_erent scalings allow for di_erent asymptotic properties of the process (small-time and tail behaviours in particular). In order to do so, we extend some results on sample path large deviations for such di_usions. As an application, we show how these results characterise the small-time and tail estimates of the implied volatility for rough volatility models, recently proposed in mathematical _nance.

17-04

Antoine Jacquier, Louis Jeannerod **How Many Paths To Simulate Correlated Brownian Motions?**

We provide an explicit formula giving the optimal number of paths needed to simulate two correlated Brownian motions.

17-05

Antoine Jacquier, Hao Liu **Optimal Liquidation In A Level-I Limit Order Book For Large-Tick Stocks**

We propose a framework to study the optimal liquidation strategy in a limit order book for large-tick stocks, with the spread equal to one tick. All order book events (market orders, limit orders and cancellations) occur according to independent Poisson processes, with parameters depending on price move directions. Our goal is to maximise the expected terminal wealth of an agent who needs to liquidate her positions within a _xed time horizon. By assuming that the agent trades (through sell limit order or/and sell market order) only when the price moves, we model her liquidation procedure as a semi-Markov decision process, and compute the semi-Markov kernel using Laplace method in the language of queueing theory. The optimal liquidation policy is then solved by dynamic programming, and illustrated numerically.

17-06

Antoine Jacquier, Mikko S. Pakkanen, Henry Stone **Pathwise Large Deviations For The Rough Bergomi Model.**

We study the small-time behaviour of the rough Bergomi model, introduced by Bayer, Friz and Gatheral [4], and prove a large deviations principle for a rescaled version of the normalised log stock price process, which then allows us to characterise the small-time behaviour of the implied volatility.

17-07

John Armstrong, Damiano Brigo

**Optimal approximation of SDEs on submanifolds: the Ito-vector and Ito-jet projections.**

We define two new notions of projection of a stochastic differential equation (SDE) onto a submanifold: the Ito-vector and Ito-jet projections. This allows one to systematically develop low dimensional approximations to high dimensional SDEs using differential geometric techniques. The approach generalizes the notion of projecting a vector field onto a submanifold in order to derive approximations to ordinary differential equations, and improves the previous Stratonovich projection method by adding optimality analysis and results. Indeed, just as in the case of ordinary projection, our definitions of projection are based on optimality arguments and give in a well-defined sense "optimal" approximations to the original SDE in the mean-square sense. We also show that the Stratonovich projection satisfies an optimality criterion that is more ad hoc and less appealing than the criteria satisfied by the Ito projections we introduce. As an application we consider approximating the solution of the non-linear filtering problem with a Gaussian distribution and show how the newly introduced Ito projections lead to optimal approximations in the Gaussian family and briefly discuss the optimal approximation for more general families of distribution. We perform a numerical comparison of our optimally approximated filter with the classical Extended Kalman Filter to demonstrate the efficacy of the approach.

17-08

Damiano Brigo, Giovanni Pistone

**Optimal approximations of the Fokker-Planck-Kolmogorov equation: projection, maximum likelihood eigenfunctions and Galerkin methods.**

We study optimal finite dimensional approximations of the generally infinite-dimensional Fokker-Planck-Kolmogorov (FPK) equation, finding the curve in a given finite-dimensional family that best approximates the exact solution evolution. For a first local approximation we assign a manifold structure to the family and a metric. We then project the vector field of the partial differential equation (PDE) onto the tangent space of the chosen family, thus obtaining an ordinary differential equation for the family parameter. A second global approximation will be based on projecting directly the exact solution from its infinite dimensional space to the chosen family using the nonlinear metric projection. This will result in matching expectations with respect to the exact and approximating densities for particular functions associated with the chosen family, but this will require knowledge of the exact solution of FPK. A first way around this is a localized version of the metric projection based on the assumed density approximation. While the localization will remove global optimality, we will show that the somewhat arbitrary assumed density approximation is equivalent to the mathematically rigorous vector field projection. More interestingly we study the case where the approximating family is defined based on a number of eigenfunctions of the exact equation. In this case we show that the local vector field projection provides also the globally optimal approximation in metric projection, and for some families this coincides with a Galerkin method.

17-09

Francesc Pons Llopis , Nikolas Kantas, Alexandros Beskosy, Ajay Jasra

**Particle Filtering for Stochastic Navier-Stokes Signal Observed with Linear Additive Noise.**

We consider a non-linear filtering problem, whereby the signal obeys the stochastic Navier- Stokes equations and is observed through a linear mapping with additive noise. The setup is relevant to data assimilation for numerical weather prediction and climate modelling, where similar models are used for unknown ocean or wind velocities. We present a particle filtering methodology that uses likelihood informed importance proposals, adaptive tempering, and a small number of appropriate Markov Chain Monte Carlo steps. We provide a detailed design for each of these steps and show in our numerical examples that they are all crucial in terms of achieving good performance and efficiency.

17-10

Alexander Kalinin, Alexander Schied**Mild and viscosity solutions to semilinear parabolic path-dependent PDEs.**

We study and compare two concepts for weak solutions to semilinear parabolic path-dependent partial differential equations (PPDEs). The first is that of mild solutions as it appears, e.g., in the log-Laplace functionals of historical superprocesses. The aim of this paper is to show that mild solutions are also solutions in a viscosity sense. This result is motivated by the fact that mild solutions can provide value functions and optimal strategies for problems of stochastic optimal control. Since unique mild solutions exist under weak conditions, we obtain as a corollary a general existence result for viscosity solutions to semiilinear parabolic PPDEs.

17-11

Thomas Cass, Nengli Lim**A Stratonovich-Skorohod integral formula for Gaussian rough paths.**

Given a Gaussian process X , its canonical geometric rough path lift X , and a solution Y to the rough differential equation (RDE) dY t =V(Y t )∘dX t , we present a closed-form correction formula for ∫Y∘dX−∫YdX , i.e. the difference between the rough and Skorohod integrals of Y with respect to X . When X is standard Brownian motion, we recover the classical Stratonovich-to-It{\^o} conversion formula, which we generalize to Gaussian rough paths with finite p -variation, p<3 , and satisfying an additional natural condition. This encompasses many familiar examples, including fractional Brownian motion with H>13 . To prove the formula, we first show that the Riemann-sum approximants of the Skorohod integral converge in L 2 (Ω) by using a novel characterization of the Cameron-Martin norm in terms of higher-dimensional Young-Stieltjes integrals. Next, we append the approximants of the Skorohod integral with a suitable compensation term without altering the limit, and the formula is finally obtained after a re-balancing of terms.

17-12

Thomas Cass, Martin P. Weidner**Tree algebras over topological vector spaces in rough path theory.**

We work with non-planar rooted trees which have a label set given by an arbitrary vector space V . By equipping V with a complete locally convex topology, we show how a natural topology is induced on the tree algebra over V . In this context, we introduce the Grossman-Larson and Connes-Kreimer topological Hopf algebras over V , and prove that they form a dual pair in a certain sense. As an application we define the class of branched rough paths over a general Banach space, and propose a new definition of a solution to a rough differential equation (RDE) driven by one of these branched rough paths. We show equivalence of our definition with a Davie-Friz-Victoir-type definition, a version of which is widely used for RDEs with geometric drivers, and we comment on applications to RDEs with manifold-valued solutions.

17-13

Dan Crisan, Franco Flandoli, Darryl D. Holm**Solution properties of a 3D stochastic Euler fluid equation.**

We prove local well posedness in regular spaces and a Beale-Kato-Majda blow-up criterion for a recently derived stochastic model of the 3D Euler fluid equation for incompressible flow. This model describes incompressible fluid motions whose Lagrangian particle paths follow a stochastic process with cylindrical noise and also satisfy Newton's 2nd Law in every Lagrangian domain.

17-14

Julien Barré, Dan Crisan, Thierry Goudon**Two-dimensional pseudo-gravity model.**

We analyze a simple macroscopic model describing the evolution of a cloud of particles confined in a magneto-optical trap. The behavior of the particles is mainly driven by self--consistent attractive forces. In contrast to the standard model of gravitational forces, the force field does not result from a potential; moreover, the non linear coupling is more singular than the coupling based on the Poisson equation. We establish the existence of solutions, under a suitable smallness condition on the total mass, or, equivalently, for a sufficiently large diffusion coefficient. When a symmetry assumption is fulfilled, the solutions satisfy strengthened estimates (exponential moments). We also investigate the convergence of the N -particles description towards the PDE system in the mean field regime.

17-15

Martin Hairer, Cyril Labbé**The reconstruction theorem in Besov spaces.**

The theory of regularity structures sets up an abstract framework of modelled distributions generalising the usual H\"older functions and allowing one to give a meaning to several ill-posed stochastic PDEs. A key result in that theory is the so-called reconstruction theorem: it defines a continuous linear operator that maps spaces of "modelled distributions" into the usual space of distributions. In the present paper, we extend the scope of this theorem to analogues to the whole class of Besov spaces B γ p,q with non-integer regularity indices. We then show that these spaces behave very much like their classical counterparts by obtaining the corresponding embedding theorems and Schauder-type estimates.

17-16

Martin Hairer, Jonathan Mattingly**The strong Feller property for singular stochastic PDEs.**

We show that the Markov semigroups generated by a large class of singular stochastic PDEs satisfy the strong Feller property. These include for example the KPZ equation and the dynamical Φ 4 3 model. As a corollary, we prove that the Brownian bridge measure is the unique invariant measure for the KPZ equation with periodic boundary conditions.

17-17

C. Bayer, P. K. Friz, A. Gulisashvili, B. Horvath, B. Stemper

Short-Time Near-The-Money Skew In Roughfractional Volatility Models.

We consider rough stochastic volatility models where the driving noise of volatility has fractional scaling, in the \rough" regime of Hurst parameter H < 1=2. This regime recently attracted a lot of attention both from the statistical and option pricing point of view. With focus on the latter, we sharpen the large deviation results of Forde-Zhang (2017) in a way that allows us to zoom-in around the money while maintaining full analytical tractability. More precisely, this amounts to proving higher order moderate deviation estimates, only recently introduced in the option pricing context. This in turn allows us to push the applicability range of known at-the-money skew approximation formulae from CLT type log-moneyness deviations of order t1=2 (recent works of Alos, Leon & Vives and Fukasawa) to the wider moderate deviations regime.

17-18

Francesc Pons Llopis, Nikolas Kantas, Alexandros Beskos, Ajay Jasra**Particle Filtering for Stochastic Navier-Stokes Signal Observed with Linear Additive Noise.**

We consider a non-linear filtering problem, whereby the signal obeys the stochastic Navier-Stokes equations and is observed through a linear mapping with additive noise. The setup is relevant to data assimilation for numerical weather prediction and climate modelling, where similar models are used for unknown ocean or wind velocities. We present a particle filtering methodology that uses likelihood informed importance proposals, adaptive tempering, and a small number of appropriate Markov Chain Monte Carlo steps. We provide a detailed design for each of these steps and show in our numerical examples that they are all crucial in terms of achieving good performance and efficiency.

17-19

Miguel A. Durán-Olivencia, Rishabh S. Gvalani, Serafim Kalliadasis, Grigorios A. Pavliotis**Instability, rupture and fluctuations in thin liquid films: Theory and computations.**

Thin liquid films are ubiquitous in natural phenomena and technological applications. They have been extensively studied via deterministic hydrodynamic equations, but thermal fluctuations often play a crucial role that needs to be understood. An example of this is dewetting, which involves the rupture of a thin liquid film and the formation of droplets. Such a process is thermally activated and requires fluctuations to be taken into account self-consistently. In this work we present an analytical and numerical study of a stochastic thin-film equation derived from first principles. Following a brief review of the derivation, we scrutinise the behaviour of the equation in the limit of perfectly correlated noise along the wall-normal direction. The stochastic thin-film equation is also simulated by adopting a numerical scheme based on a spectral collocation method. The scheme allows us to explore the fluctuating dynamics of the thin film and the behaviour of its free energy in the vicinity of rupture. Finally, we also study the effect of the noise intensity on the rupture time, which is in agreement with previous works.

17-20

S. N. Gomes, G. A. Pavliotis**Mean Field Limits for Interacting Diffusions in a Two-Scale Potential.**

In this paper we study the combined mean field and homogenization limits for a system of weakly interacting diffusions moving in a two-scale, locally periodic confining potential, of the form considered in~\cite{DuncanPavliotis2016}. We show that, although the mean field and homogenization limits commute for finite times, they do not, in general, commute in the long time limit. In particular, the bifurcation diagrams for the stationary states can be different depending on the order with which we take the two limits. Furthermore, we construct the bifurcation diagram for the stationary McKean-Vlasov equation in a two-scale potential, before passing to the homogenization limit, and we analyze the effect of the multiple local minima in the confining potential on the number and the stability of stationary solutions.

17-21

Alexander Kalinin**Markovian Integral Equations.**

We analyze multidimensional Markovian integral equations that are formulated with a time-inhomogeneous progressive Markov process that has Borel measurable transition probabilities. In the case of a path-dependent diffusion process, the solutions to these integral equations lead to the concept of mild solutions to semilinear parabolic path-dependent partial differential equations (PPDEs). Our goal is to establish uniqueness, stability, existence, and non-extendibility of solutions among a certain class of maps. By requiring the Feller property of the Markov process, we give weak conditions under which solutions become continuous. Moreover, we provide a multidimensional Feynman-Kac formula and a one-dimensional global existence- and uniqueness result.

17-22

Martin Hairer, Gautam Iyer, Leonid Koralov, Alexei Novikov, Zsolt Pajor-Gyulai**A fractional kinetic process describing the intermediate time behaviour of cellular flows.**

This paper studies the intermediate time behaviour of a small random perturbation of a periodic cellular flow. Our main result shows that on time scales shorter than the diffusive time scale, the limiting behaviour of trajectories that start close enough to cell boundaries is a fractional kinetic process: A Brownian motion time changed by the local time of an independent Brownian motion. Our proof uses the Freidlin-Wentzell framework, and the key step is to establish an analogous averaging principle on shorter time scales. As a consequence of our main theorem, we obtain a homogenization result for the associated advection-diffusion equation. We show that on intermediate time scales the effective equation is a fractional time PDE that arises in modelling anomalous diffusion.

17-23

Xue-Mei Li**Perturbation of Conservation Laws and Averaging on Manifolds.**

We prove a stochastic averaging theorem for stochastic differential equationsinwhichtheslowandthefastvariablesinteract.TheapproximateMarkovfast motionisafamilyofMarkovprocesswithgenerator Lx.Thetheoremisprovedunder the assumption that Lx satisﬁes H¨ormander’s bracket conditions, or more generallyLx isafamilyofFredholmoperatorswithsub-ellipticestimates.Ontheother hand a conservation law of a dynamical system can be used as a tool for separating the scales in singular perturbation problems. We discuss a number of motivating examples from mathematical physics and from geometry where we use non-linear conservation laws to deduce slow-fast systems of stochastic differential equations.

17-24

Xue-Mei Li**Doubly Damped Stochastic Parallel Translations and Hessian Formulas.**

We study the Hessian of the solutions of time-independent Schr¨odinger equations, aiming to obtain as large a class as possible of complete Riemannian manifoldsforwhichtheestimateC(1 t + d2 t2 )holds.Forthispurposeweintroducethe doubly damped stochastic parallel transport equation, study them and make exponential estimates on them, deduce a second order Feynman-Kac formula and obtain the desired estimates. Our aim here is to explain the intuition, the basic techniques, and the formulas which might be useful in other studies. AMS subject classiﬁcation. 60Gxx, 60Hxx, 58J65, 58J70. Keywords. Heat kernels, weighted Laplacian, Schr¨odinger operators, Hessian formulas, Hessian estimates.

17-25

Alejandro Gomez, Jong Jun Lee, Carl Mueller, Eyal Neuman, Michael Salins**On uniqueness and blowup properties for a class of second order SDEs.**

As the ﬁrst step for approaching the uniqueness and blowup properties of the solutions of the stochastic wave equations with multiplicative noise, we analyze the conditions for the uniqueness and blowup properties of the solution (Xt,Yt) of the equations dXt = Ytdt, dYt = |Xt|αdBt, (X0,Y0) = (x0,y0). In particular, we prove that solutions are nonunique if 0 < α < 1 and (x0,y0) = (0,0) and unique if 1/2 < α and (x0,y0) 6= (0,0). We also show that blowup in ﬁnite time holds if α > 1 and (x0,y0) 6= (0,0).

17-26**On pinned fields, interlacements, and random walk on (Z/NZ)2**

Pierre-François Rodriguez

We define two families of Poissonian soups of bidirectional trajectories on Z2, which can be seen to adequately describe the local picture of the trace left by a random walk on the two-dimensional torus (Z/NZ)2, started from the uniform distribution, run up to a time of order (NlogN)2 and forced to avoid a fixed point. The local limit of the latter was recently established in arXiv:1502.03470. Our construction proceeds by considering, somewhat in the spirit of statistical mechanics, a sequence of finite-volume approximations, consisting of random walks avoiding the origin and killed at spatial scale N, either using Dirichlet boundary conditions, or by means of a suitably adjusted mass. By tuning the intensity u of such walks with N, the occupation field can be seen to have a nontrivial limit, corresponding to that of the actual random walk. Our construction thus yields a two-dimensional analogue of the random interlacements model introduced in arXiv:0704.2560 in the transient case. It also links it to the pinned free field in Z2, by means of a (pinned) Ray-Knight type isomorphism theorem.

### 2016

Sergey Badikov, Antoine Jacquier, Daphne Qing Liu, Patrick Roome

**No-arbitrage bounds for the forward smile given marginals**

We explore the robust replication of forward-start straddles given quoted (Call and Put options) market data. One approach to this problem classically follows semi-infinite linear programming arguments, and we propose a discretisation scheme to reduce its dimensionality and hence its complexity. Alternatively, one can consider the dual problem, consisting in finding optimal martingale measures under which the upper and the lower bounds are attained. Semi-analytical solutions to this dual problem were proposed by Hobson and Klimmek [13] and by Hobson and Neuberger [14]. We recast this dual approach as a finite dimensional linear programme, and reconcile numerically, in the Black-Scholes and in the Heston model, the two approaches.

16-02

Thomas Cass, Nengli Lim**A Stratonovich-Skorohod integral formula for Gaussian rough paths**

Given a Gaussian process X, its canonical geometric rough path lift X, and a solution Y to the rough differential equation (RDE) dYt=V(Yt)∘dXt, we present a closed-form correction formula for ∫Y∘dX−∫YdX, i.e. the difference between the rough and Skorohod integrals of Y with respect to X. When X is standard Brownian motion, we recover the classical Stratonovich-to-It\^o conversion formula, which we generalize to Gaussian rough paths with finite p-variation, 2<p<3, and satisfying an additional natural condition. This encompasses many familiar examples, including fractional Brownian motion with H>13. To prove the formula, we show that ∫YdX is the L2(Ω) limit of its Riemann-sum approximants, and that the approximants can be appended with a suitable compensation term without altering the limit. To show convergence of the Riemann-sum approximants, we utilize a novel characterization of the Cameron-Martin norm using higher-dimensional Young-Stieltjes integrals. For the main theorem, complementary regularity between the Cameron-Martin paths and the covariance function of X is used to show the existence of these integrals. However, it turns out not to be a necessary condition, as in the last section we provide a new set of conditions for their existence.

16-03

Thomas Cass, Martin P. Weidner**Hörmander's theorem for rough differential equations on manifolds**

We introduce a new definition for solutions Y to rough differential equations (RDEs) of the form dYt=V(Yt)dXt,Y0=y0. By using the Grossman-Larson Hopf algebra on labelled rooted trees, we prove equivalence with the classical definition of a solution advanced by Davie when the state space E for Y is a finite-dimensional vector space. The notion of solution we propose, however, works when E is any smooth manifold M and is therefore equally well-suited for use as an intrinsic defintion of an M-valued RDE solution. This enables us to prove existence, uniqueness and coordinate-invariant theorems for RDEs on M bypassing the need to define a rough path on M. Using this framework, we generalise result of Cass, Hairer, Litter and Tindel proving the smoothness of the density of M-valued RDEs driven by non-degenerate Gaussian rough paths under Hörmander's** **bracket condition. In doing so, we reinterpret some of the foundational results of the Malliavin calculus to make them appropriate to the study M-valued Wiener functionals.

John Armstrong, Damiano Brigo

**Coordinate-free Stochastic Differential Equations as Jets**

We explain how Itô Stochastic Dierential Equations on manifolds may be dened as 2-jets of curves. We use jets as a natural language to express geometric properties of SDEs and show how jets can lead to intuitive representations of Itô SDEs, including three dierent types of drawings. We explain that the mainstream choice of Fisk-Stratonovich-McShane calculus for stochastic dierential geometry is not necessary and elaborate on the relationships with the jets approach. We consider the two calculi as being simply dierent coordinate systems for the same un- derlying coordinate-free stochastic dierential equation. If the extrinsic approach to dierential geometry is adopted, then Stratonovich calculus may appear to be necessary when studying SDEs on submanifolds but in fact one can use the Itô/2-jets framework proposed here by recalling that the curvature of the 2-jet follows the curvature of the manifold. We argue that the choice between Itô and Stratonovich is a modelling choice dic- tated by the type of problem one is facing and the related desiderata. We also discuss the forward Kolmogorov equation and the backward diusion operator in geometric terms, and consider percentiles of the solutions of the SDE and their properties, leading to fan diagrams and their relation- ship with jets. In particular, the median of a SDE solution is associated to the drift of the SDE in Stratonovich form for small times. Finally, we prove convergence of the 2-jet scheme to classical Itô SDEs solutions.

Damiano Brigo, Marco Francischello, Andrea Pallavicini

**Invariance, existence and uniqueness of solutions of nonlinear valuation PDEs and FBSDEs inclusive of credit risk, collateral and funding costs**

We study conditions for existence, uniqueness and invariance of the comprehensive nonlinear valuation equations first introduced in Pallavicini et al (2011) [11]. These equations take the form of semi-linear PDEs and Forward-Backward Stochastic Differential Equations (FBSDEs). After summarizing the cash flows definitions allowing us to extend valuation to credit risk and default closeout, including collateral margining with possible re-hypothecation, and treasury funding costs, we show how such cash flows, when present-valued in an arbitrage free setting, lead to semi-linear PDEs or more generally to FBSDEs. We provide conditions for existence and uniqueness of such solutions in a viscosity and classical sense, discussing the role of the hedging strategy. We show an invariance theorem stating that even though we start from a risk-neutral valuation approach based on a locally risk-free bank account growing at a risk-free rate, our final valuation equations do not depend on the risk free rate. Indeed, our final semilinear PDE or FBSDEs and their classical or viscosity solutions depend only on contractual, market or treasury rates and we do not need to proxy the risk free rate with a real market rate, since it acts as an instrumental variable. The equations derivations, their numerical solutions, the related XVA valuation adjustments with their overlap, and the invariance result had been analyzed numerically and extended to central clearing and multiple discount curves in a number of previous works, including [11], [12], [10], [6] and [4].

Damiano Brigo, Giovanni Pistone

**Projection based dimensionality reduction for** **measure valued ****evolution equations in statistical manifolds**

We propose a dimensionality reduction method for infinite-dimensional measure-valued evolution equations such as the Fokker-Planck partial differential equation or the Kushner-Stratonovich resp. Duncan-Mortensen-Zakai stochastic partial differential equations of nonlinear filtering, with potential applications to signal processing, quantitative finance, heat flows and quantum theory among many other areas. Our method is based on the projection coming from a duality argument built in the exponential statistical manifold structure developed by G. Pistone and co-authors. The choice of the finite dimensional manifold on which one should project the infinite dimensional equation is crucial, and we propose finite dimensional exponential and mixture families. This same problem had been studied, especially in the context of nonlinear filtering, by D. Brigo and co-authors but the L2 structure on the space of square roots of densities or of densities themselves was used, without taking an infinite dimensional manifold environment space for the equation to be projected. Here we re-examine such works from the exponential statistical manifold point of view, which allows for a deeper geometric understanding of the manifold structures at play. We also show that the projection in the exponential manifold structure is

consistent with the Fisher Rao metric and, in case of finite dimensional exponential families, with the assumed density approximation. Further, we show that if the

sufficient statistics of the finite dimensional exponential family are chosen among the eigenfunctions of the backward diffusion operator then the statistical-manifold

or Fisher-Rao projection provides the maximum likelihood estimator for the Fokker Planck equation solution. We finally try to clarify how the nite dimensional and infinite dimensional terminology for exponential and mixture spaces are related.

Damiano Brigo, Giovanni Pistone

**Maximum likelihood eigenfunctions of the Fokker Planck equation and Hellinger projection**

We apply the L2 based Fisher-Rao vector-field projection by Brigo, Hanzon and LeGland (1999) to finite dimensional approximations of the Fokker Planck equation on exponential families. We show that if the sufficient statistics are chosen among the diffusion eigenfunctions the finite dimensional projection or the equivalent assumed density approximation provide the exact maximum likelihood density. The same result had been derived earlier by Brigo and Pistone (2016) in the infinite-dimensional Orlicz based geometry of Pistone and co-authors as opposed to the L2 structure used here.

**The Local Fractional Bootstrap**

Mikkel Bennedsen, Ulrich Hounyo, Asger Lunde, Mikko S. Pakkanen

We introduce a bootstrap procedure for high-frequency statistics of Brownian semistationary processes. More specifically, we focus on a hypothesis test on the roughness of sample paths of Brownian semistationary processes, which uses an estimator based on a ratio of realized power variations. Our new resampling method, the local fractional bootstrap, relies on simulating an auxiliary fractional Brownian motion that mimics the fine properties of high frequency differences of the Brownian semistationary process under the null hypothesis. We prove the first order validity of the bootstrap method and in simulations we observe that the bootstrap-based hypothesis test provides considerable finite-sample improvements over an existing test that is based on a central limit theorem. This is important when studying the roughness properties of time series data; we illustrate this by applying the bootstrap method to two empirical data sets: we assess the roughness of a time series of high-frequency asset prices and we test the validity of Kolmogorov's scaling law in atmospheric turbulence data.

**Arbitrage without borrowing or short selling?**Jani Lukkarinen, Mikko S. Pakkanen

We show that a trader, who starts with no initial wealth and is not allowed to borrow money or short sell assets, is theoretically able to attain positive wealth by continuous trading, provided that she has perfect foresight of future asset prices, given by a continuous semimartingale. Such an arbitrage strategy can be constructed as a process of finite variation that satisfies a seemingly innocuous self-financing condition, formulated using a pathwise Riemann-Stieltjes integral. Our result exemplifies the potential intricacies of formulating economically meaningful self-financing conditions in continuous time, when one leaves the conventional arbitrage-free framework.

**On the conditional small ball property of multivariate Lévy-driven moving average processes**Mikko S. Pakkanen, Tommi Sottinen, Adil Yazigi

We study whether a multivariate Lévy-driven moving average process can shadow arbitrarily closely any continuous path, starting from the present value of the process, with positive conditional probability, which we call the conditional small ball property. Our main results establish the conditional small ball property for Lévy-driven moving average processes under natural non-degeneracy conditions on the kernel function of the process and on the driving Lévy process. We discuss in depth how to verify these conditions in practice. As concrete examples, to which our results apply, we consider fractional Lévy processes and multivariate Lévy-driven Ornstein-Uhlenbeck processes.

**Joint Asymptotic Distribution of Certain Path Functionals of the Reflected Proces**Aleksandar Mijatović, Martijn Pistorius

Let τ(x) be the rst time that the re ected process Y of a Levy process X crosses x > 0. The main aim of this paper is to investigate the joint asymptotic distribution of Y (t) = X(t) inf0≤s≤t X(s) and the path functionals Z(x) = Y (τ(x)) x and m(t) = sup0≤s≤t Y (s) y∗(t), for a certain non-linear curve y∗(t). We restrict to Levy processes X satisfying Cramer's condition, a non-lattice condition and the moment conditions that E[jX(1)j] and E[exp(γX(1))jX(1)j] are nite (where γ denotes the Cramer coecient). We prove that Y (t) and Z(x) are asymptotically independent as minft, xg ! 1 and characterise the law of the limit (Y∞,Z∞). Moreover, if y∗(t) = γ−1 log(t) and minft, xg ! 1 in such a way that t expfγxg ! 0, then we show that Y (t), Z(x) and m(t) are asymptotically independent and derive the explicit form of the joint weak limit (Y∞,Z∞,m∞). The proof is based on excursion theory, Theorem 1 in [7] and our characterisation of the law (Y∞,Z∞).

**On Dynamic Deviation Measures and Continuous-Time Portfolio Optimisation**Martijn Pistorius, Mitja Stadjey

In this paper we propose the notion of dynamic deviation measure, as a dynamic timeconsistent extension of the (static) notion of deviation measure. To achieve time-consistency we require that a dynamic deviation measures satises a generalised conditional variance formula. We show that, under a domination condition, dynamic deviation measures are characterised as the solutions to a certain class of backward SDEs. We establish for any dynamic deviation measure an integral representation, and derive a dual characterisation result in terms of additively m-stable dual sets. Using this notion of dynamic deviation measure we formulate a dynamic mean-deviation portfolio optimisation problem in a jump-diusion setting and identify a subgame-perfect Nash equilibrium strategy that is linear as function of wealth by deriving and solving an associated extended HJB equation.

### 2015

**Working Papers 2015**

15-01

Archil Gulisashvili, Blanka Horvath & Antoine Jacquier**Mass at Zero and Small-Strike Implied Volatility Expansion in the Sabr Model.**

*We study the probability mass at the origin in the SABR stochastic volatility model, and derive several tractable expressions for it, in particular when time becomes small or large. In the uncorrelated case, tedious saddlepoint expansions allow for (semi) closed-form asymp-totic formulae. As an application–the original motivation for this paper–we derive small-strike expansions for the implied volatility when the maturity becomes short or large. These formulae, by definition arbitrage-free, allow us to quantify the impact of the mass at zero on currently used implied volatility expansions. In particular we discuss how much those expansions become erroneous.*

15-02

Aleksandar Mijatovic, Martijn Pistorius*Buffer-Overflows: Joint Limit Laws of Undershoots and Overshoots of Reflected Processes.*

15-03

Thomas Cass, Bruce K. Drivery, Christian Littererz**Constrained Rough Paths.**

*Abstract: We introduce a notion of rough paths on embedded submanifolds and demonstrate that this class of rough paths is natural. On the way we develop a notion of rough integration and an ecient and intrinsic theory of rough di erential equations (RDEs) on manifolds. The theory of RDEs is then used to construct parallel translation along manifold valued rough paths. Finally, this framework is used to show there is a one to one correspondence between rough paths on a d { dimensional manifold and rough paths on d { dimensional Euclidean space. This last result is a rough path analogue of Cartan's development map and its stochastic version which was developed by Eells and Elworthy and Malliavin.*

15-04

Thomas Cass, Marcel Ogrodnik*Tail estimates for Markovian rough paths.*

*Abstract: We work in the context of Markovian rough paths associated to a class of uniformly subelliptic Dirichlet forms ([25]) and prove an almost-Gaussian tail- estimate for the accummulated local p-variation functional, which has been intro- duced and studied in [17]. We comment on the signi cance of these estimates to a range of currently-studied problems, including the recent results of Chevyrev and Lyons in [18].*

*To appear in Annals of Probability*

15-05

Antoine Jacquier, Partick Roome*Black-Scholes in a CEV random environment: a new approach to smile modelling.*

*Classical (Itˆo diffusions) stochastic volatility models are not able to capture the steepness of small-maturity implied volatility smiles. Jumps, in particular exponential Levy and affine models, which exhibit small-maturity exploding smiles, have historically been proposed to remedy this (see [53] for an overview). A recent breakthrough was made by Gatheral, Jaisson and Rosenbaum [27], who proposed to replace the Brownian driver of the instantaneous volatility by a short-memory fractional Brownian motion, which is able to capture the short-maturity steepness while preserving path continuity. We suggest here a different route, randomising the Black-Scholes variance by a CEV-generated distribution, which allows us to modulate the rate of explosion (through the CEV exponent) of the implied volatility for small maturities. The range of rates includes behaviours similar to exponential Levy models and fractional stochastic volatility models. As a by-product, we make a conjecture on the small-maturity forward smile asymptotic s of stochastic volatility models, in exact agreement with the results in [37] for the Heston model.*

15-06** **Mikkel Bennedsen, Asger Lundey, Mikko S. Pakkanenz

Hybrid scheme for Brownian semistationary processes

Hybrid scheme for Brownian semistationary processes

*We introduce a simulation scheme for Brownian semistationary processes, which is based* *on discretizing the stochastic integral representation of the process in the time domain. We* *assume that the kernel function of the process is regularly varying at zero. The novel feature* *of the scheme is to approximate the kernel function by a power function near zero and by a* *step function elsewhere. The resulting approximation of the process is a combination of Wiener* *integrals of the power function and a Riemann sum, which is why we call this method a hybrid* *scheme. Our main theoretical result describes the asymptotics of the mean square error of the* *hybrid scheme and we observe that the scheme leads to a substantial improvement of accuracy* *compared to the ordinary forward Riemann-sum scheme, while having the same computational**complexity. We exemplify the use of the hybrid scheme by two numerical experiments, where we* *examine the nite-sample properties of an estimator of the roughness parameter of a Brownian* *semistationary process and study Monte Carlo option pricing in the rough Bergomi model of* **Bayer et al. (2015)**, respectively.

### 2014

**Working Papers 2014**

14-01

Jean-Francois Chassagneux, Antoine Jacquier, Ivo Mihaylov**An explicit Euler scheme with strong rate of convergence for non-Lipschitz SDEs**

*Abstract: We consider the approximation of stochastic differential equations (SDEs) with non-Lipschitz drift or diffusion coefficients. We present a modified explicit Euler-Maruyama discretisation scheme that allows us to prove strong convergence, with a rate. Under some regularity conditions, we obtain the optimal strong error rate. We consider SDEs popular in the mathematical finance literature, including the Cox-Ingersoll-Ross (CIR), the 3=2 and the Ait-Sahalia models, as well as a family of mean-reverting processes with locally smooth coefficients.*

*Keywords: Stochastic differential equations, non-Lipschitz coefficients, explicit Euler-Maruyama scheme with projection, CIR model, Ait-Sahalia model.*

14-02

Antoine Jacquier and Patrick Roome *Large-maturity regimes of the Heston Forward Smile*

*Abstract: We provide a full characterisation of the large-maturity forward implied volatility smile in the Heston model. Although the leading decay is provided by a fairly classical large deviations behaviour, the algebraic expansion providing the higher-order terms highly depends on the parameters, and di erent powers of the maturity come into play. As a by-product of the analysis we provide new implied volatility asymptotics, both in the forward case and in the spot case, as well as extended SVI-type formulae. The proofs are based on extensions and re nements of sharp large deviations theory, in particular in cases where standard convexity arguments fail.*

14-03

Jean- François Chassagneux and Adrien Richou *Numerical Stability Analysis of the Euler Scheme for BSDES*

*Abstract: In this paper, we study the qualitative behaviour of approximation schemes for Backward Stochastic Differential Equations (BSDEs) by introducing a new notion of numerical stability. For the Euler scheme, we provide sufficient conditions in the one-dimensional and multidimensional case to guarantee the numerical stability. We then perform a classical Von Neumann stability analysis in the case of a linear driver f and exhibit necessary conditions to get stability in this case. Finally, we illustrate our results with numerical applications.*

14-04

M. Ottobre, G.A. Pavlotis, K. Pravda-Starov *Some remarks on Degenrate Hypoelliptic Ornstein-Uhlenbeck Operators*

*Abstract: We study degenerate hypoelliptic Ornstein-Uhlenbeck operators in L2 spaces with respect to invariant measures. The purpose of this article is to show how recent results on general quadratic operators apply to the study of degenerate hypoelliptic Ornstein-Uhlenbeck operators. We rst show that some known results about the spectral and subelliptic properties of Ornstein-Uhlenbeck operators may be directly recovered from the general analysis of quadratic operators with zero singular spaces. We also provide new resolvent estimates for hypoelliptic Ornstein-Uhlenbeck operators. We show in particular that the spectrum of these non-selfadjoint operators may be very unstable under small perturbations and that their resolvents can blow-up in norm far away from their spectra. Furthermore, we establish sharp resolvent estimates in speci c regions of the resolvent set which enable us to prove exponential return to equilibrium.*

14-05

John Armstrong, Damiano Brigo*Stochastic Filtering via L2 projection on mixture manifolds with computer algorithms and numerical examples*

*Abstract: We examine some differential geometric approaches to finding approximate solutions to the continuous time nonlinear filtering problem. Our primary focus is a new projection method for the optimal filter infinite dimensional Stochastic Partial Differential Equation (SPDE), based on the direct L2 metric and on a family of normal mixtures. We compare this method to earlier projection methods based on the Hellinger distance/Fisher metric and exponential families, and we compare the L2 mixture projection filter with a particle method with the same number of parameters, using the Levy metric. We prove that for a simple choice of the mixture manifold the L2 mixture projection filter coincides with a Galerkin method, whereas for more general mixture manifolds the equivalence does not hold and the L2 mixture filter is more general. We study particular systems that may illustrate the advantages of this new filter over other algorithms when comparing outputs with the optimal filter. We finally consider a specific software design that is suited for a numerically efficient implementation of this filter and provide numerical examples.*

*Keywords: Direct L2 metric, Exponential Families, Finite Dimensional, Families of Probability Distributions, Fisher information metric, Hellinger distance, Levy Metric, Mixture Families, Stochastic filtering, Galerkin, AMS Classification codes: 53B25, 53B50, 60G35, 62E17, 62M20, 93E11*

14-06

Damiano Brigo, Jan-Frederik Mai, Matthias Scherer *Consistent iterated simulation of multi-variate default times: a Markovian indicators characterization*

*Abstract: We investigate under which conditions a single simulation of joint default times at a nal time horizon can be decomposed into a set of simulations of joint defaults on subsequent adjacent sub-periods leading to that nal horizon. Besides the theoretical interest, this is also a practical problem as part of the industry has been working under the misleading assumption that the two approaches are equivalent for practical purposes. As a reasonable trade-o between realistic stylized facts, practical demands, and mathematical tractability, we propose models leading to a Markovian multi-variate survival{indicator process, and we investigate two instances of static models for the vector of default times from the statistical literature that fall into this class. On the one hand, the "looping default" case is known to be equipped with this property at least since [Herbertsson, Rootzen (2008), Bielecki et al. (2011b)],and we point out that it coincides with the classical "Freund distribution" in the bivariate case. On the other hand, if all sub-vectors of the survival indicator process are Markovian, this constitutes a new characterization of the Marshall-Olkin distribution, and hence of multi-variate lack-of-memory. A paramount property of the resulting model is stability of the type of multi-variate distribution with respect to elimination or insertion of a new marginal component with marginal distribution from the same family. The practical implications of this "nested margining" property are enormous. To implement this distribution we present an ecient and unbiased simulation algorithm based on the Levy-frailty construction. We highlight dfferent pitfalls in the simulation of dependent default times and examine, within a numerical case study, the effect of inadequate simulation practices.*

*Keywords: Stepwise default simulation, default modeling, credit modeling, default dependence, default correlation, default simulation, arrival times, credit risk, Marshall-Olkin distribution, nested margining, Freund distribution, looping default models.*

14-07

Damiano Brigo, Francesco Rapisarday, Abir Sridiz *The arbitrage-free Multivariate Mixture Dynamics Model: Consistent single-assets and index volatility smiles*

*Abstract: We introduce a multivariate diffusion model that is able to price derivative securities featuring multiple underlying assets. Each asset volatility smile is modeled according to a density-mixture dynamical model while the same property holds for the multivariate process of all assets, whose density is a mixture of multivariate basic densities. This allowsto reconcile single name and index/basket volatility smiles in a consistent framework. Our approach could be dubbed a multidimensional local volatility approach with vector-state dependent diffusion matrix. The model is quite tractable, leading to a complete market and not requiring Fourier techniques for calibration and dependence measures, contrary to multivariate stochastic volatility models such as Wishart. We prove existence and uniqueness of solutions for the model stochastic differential equations, provide formulas for a number of basket options, and analyze the dependence structure of the model in detail by deriving a number of results on covariances, its copula function and rank correlation measures and volatilities-assets correlations. A comparison with sampling simply-correlated suitably discretized one-dimensional mixture dynamical paths is made, both in terms of option pricing and of dependence, and first order expansion relationships between the two models’ local covariances are derived. We also show existence of a multivariate uncertain volatility model of which our multivariate local volatilities model is a Markovian projection, highlighting that the projected model is smoother and avoids a number of drawbacks of the uncertain volatility version. We also show a consistency result where the Markovian projection of a geometric basket in the multivariate model is a univariate mixture dynamics model. A few numerical examples on basket and spread options pricing conclude the paper.*

* Key words: Mixture of densities, Volatility smile, Lognormal density, Multivariate local volatility, Complete Market, Option on a weighted Arithmetic average of a basket, Spread option, Option on a weighted geometric average of a basket, Markovian projection, Copula function.*

14-08

Stefano De Marco, Antoine Jacquier, Patrick Roome *Two Examples of Non Strictly Convex Large Deviations*

14-09

Dan Crisan, Yoshiki Otobe, Szymon Peszat*Inverse Problems for Stochastic transport equations*

*Abstract: Inverse problems for stochastic linear transport equations driven by a temporal or spatial white noise are discussed. We analyse stochastic linear transport equations which depend on an unknown potential and have either additive noise or multiplicative noise. We show that one can approximate the potential with arbitrary small error when the solution of the stochastic linear transport equation is observed over time at some fixed point in the state space.*

*Keywords: STOCHASTIC TRANSPORT EQUATIONS, partial observations, inverse problem.*

14-10

Dan Crisan, Christian Litterer, Terry Lyons*Kusuoka-Strook gradient bounds for the solution of the filtering equation*

*Abstract: We obtain sharp gradient bounds for perturbed di¤usion semigroups. In contrast with existing results, the perturbation is here random and the bounds obtained are pathwise. Our approach builds on the classical work of Kusuoka and Stroock [12, 14, 15, 16], and extends their program developed for the heat semi-group to solutions of stochastic partial di¤erential equations. The work is motivated by and applied to nonlinear ...ltering. The analysis allows us to derive pathwise gradient bounds for the un-normalised conditional distribution of a partially observed signal. It uses a pathwise representation of the per-turbed semigroup following Ocone [21]. The estimates we derive have sharp small time asymptotics.*

*Keywords: Stochastic partial di¤erential equation; Filtering; Zakai equation; Ran- domly perturbed semigroup, gradient bounds, small time asymptotics.*

14-11

Dilip Madan, Martijn Pistorius, Mitja Stadje*Convergence of BSΔEs driven by random walks to BSDES: the caseof (in)finite activity jumps with general driver*

*Abstract: In this paper we present a weak approximation scheme for BSΔEs driven by a Wiener process and an (in) nite activity Poisson random measure with drivers that are general Lipschitz functionals of the solution of the BSDE. The approximating backward stochastic difference equations (BSΔEs) are driven by random walks that weakly approximate the given Wiener process and Poisson random measure. We establish the weak convergence to the solution of the BSDE and the numerical stability of the sequence of solutions of the BSΔEs. By way of illustration we analyse explicitly a scheme with discrete step-size distributions.*

14-12

Zbigniew Michna, Zbigniew Palmowski, and Martijn Pistorius*The distribution of the supremum for spectrally asymmetric L´evy processes*

*Abstract: In this article we derive formulas for the probability IP(suptT X(t) > u) T > 0 and IP(supt<1 X(t) > u) where X is a spectrally positive L´evy process with infinite variation. The formulas are generalizations of the well-known Tak´acs formulas for stochastic processes with non-negative and interchangeable increments. Moreover, we find the joint distribution of inftT Y (t) and Y (T) where Y is a spectrally negative L´evy process.*

*Keywords: L´evy process, distribution of the supremum of a stochastic process, spectrally asymmetric L´evy process*

### 2013

#### Working Papers 2013

13-01

F. Avram, Z. Palmowski, M. Pistorius*On Gerber-Shiu functions and optimal dividend distribution for a Levy risk-process in the presence of a penalty function*

*Abstract: In this paper we consider an optimal dividend problem for an insurance company which risk process evolves as a spectrally negative Levy process (in the absence of dividend payments). We assume that the management of the company controls timing and size of dividend payments. The objective is to maximize the sum of the expected cumulative discounted dividends received until the moment of ruin and a penalty payment at the moment of ruin which is an increasing function of the size of the shortfall at ruin; in addition, there may be a fixed cost for taking out dividends. We explicitly solve the corresponding optimal control problem. The solution rests on the characterization of the value-function as (i) the unique stochastic solution of the associated HJB equation and as (ii) the pointwise smallest stochastic supersolution. We show that the optimal value process admits a dividend-penalty decomposition as sum of a martingale (associated to the penalty payment at ruin) and a potential (associated to the dividend payments). We find also an explicit necessary and sufficient condition for optimality of a single dividend-band strategy, in terms of a particular Gerber-Shiu function. We analyze a number of concrete examples.*

*Keywords: Optimal control, L´evy process, De Finetti model, transaction costs, singular control, variational inequality, barrier policies, band policies, Gerber-Shiu function.*

13-02

D. Madan, M. Pistorius, M. Stadje*On consistent valuations based on distorted expectations: from multinomial random walks to L'{e}vy processes*

*Abstract: A distorted expectation is a Choquet expectation with respect to the capacity induced by a concave probability distortion. Distorted expectations are encountered in various static settings, in risk theory, mathematical finance and mathematical economics. There are a number of different ways to extend a distorted expectation to a multi-period setting, which are not all time-consistent. One time-consistent extension is to define the non-linear expectation by backward recursion, applying the distorted expectation stepwise, over single periods. In a multinomial random walk model we show that this non-linear expectation is stable when the number of intermediate periods increases to infinity: Under a suitable scaling of the probability distortions and provided that the tick-size and time step-size converge to zero in such a way that the multinomial random walks converge to a Levy process, we show that values of random variables under the multi-period distorted expectations converge to the values under a continuous-time non-linear expectation operator, which may be identified with a certain type of Peng's g-expectation. A coupling argument is given to show that this operator reduces to a classical linear expectation when restricted to the set of pathwise increasing claims. Our results also show that a certain class of g-expectations driven by a Brownian motion and a Poisson random measure may be computed numerically by recursively defined distorted expectations.*

*Keywords: g-expectation, non-linear expectation, probability distortion, option pricing, risk measurement, convergence, L?evy process, multinomial tree.*

13-03

A. Mijatovic, M. Urusov*On the loss of the semimartingale property at the hitting time of a level*

*Abstract: This paper studies the loss of the semimartingale property of the process $g(Y)$ at the time a one-dimensional diffusion $Y$ hits a level, where $g$ is a difference of two convex functions. We show that the process $g(Y)$ can fail to be a semimartingale in two ways only, which leads to a natural definition of non-semimartingales of the \textit{first} and \textit{second kind}. We give a deterministic if and only if condition (in terms of $g$ and the coefficients of $Y$) for $g(Y)$ to fall into one of the two classes of processes, which yields a characterisation for the loss of the semimartingale property. A number of applications of the results in the theory of stochastic processes and real analysis are given: e.g. we construct an adapted diffusion $Y$ on $[0,\infty)$ and a \emph{predictable} finite stopping time $\zeta$, such that $Y$ is a semimartingale on the stochastic interval $[0,\zeta)$, continuous at $\zeta$ and constant after $\zeta$, but is \emph{not} a semimartingale on $[0,\infty)$.*

*Keywords: Continuous semimartingale; one-dimensional diffusion; local time; additive functional; Ray-Knight theorem.*

13-04

D. Crisan, S. Ortiz-Latorrey *A Kusuoka-Lyons-Victoir particle filter*

*Abstract: The aim of this paper is to introduce a new numerical algorithm for solving the continuous time non-linear ltering problem. In particular, we present a particle lter that combines the Kusuoka-Lyons-Victoir cubature method on Wiener space (KLV) [13], [18] to approximate the law of the signal with a minimal variance "thining" method, called the tree based branching algorithm (TBBA) to keep the size of the cubature tree constant in time. The novelty of our approach resides in the adaptation of the TBBA algorithm to simultaneously control the computational effort and incorporate the observation data into the system. We provide the rate of convergence of the approximating particle lter in terms of the computational effort (number of particles) and the discretization grid mesh. Finally, we test the performance of the new algorithm on a benchmark problem (the Bene?s filter).*

*Keywords: Cubature on Wiener space; particle filters; TBBA.*

13-05

A. Jacquier, P. Roome *The Small-Maturity Heston Forward Smile*

*Abstract: In this paper we investigate the asymptotics of forward-start options and the forward implied volatility smile in the Heston model as the maturity approaches zero. We prove that the forward smile for out-ofthe- money options explodes and compute a closed-form high-order expansion detailing the rate of the explosion. Furthermore the result shows that the square-root behaviour of the variance process induces a singularity such that for certain parameter con gurations one cannot obtain high-order out-of-the-money forward smile asymptotics. In the at-the-money case a separate model-independent analysis shows that the small-maturity limit is well de ned for any It^o di usion. The proofs rely on the theory of sharp large deviations (and re nements) and incidentally we provide an example of degenerate large deviations behaviour.*

*Keywords: Stochastic volatility model, Heston model, forward implied volatilty, asymptotic expansion.*

13-06

F. Haba, A. Jacquier *Asymptotic arbitrage in the Heston model*

*Abstract: In the context of the Heston model, we establish a precise link between the set of equivalent martingale measures, the ergodicity of the underlying variance process and the concept of asymptotic arbitrage proposed in Kabanov-Kramkov [13] and in Follmer-Schachermayer [8].*

*Keywords: Stochastic volatility model, Heston model, asymptotic arbitrage, large deviations.*

13-07

S. Jacka, A. Mijatovic, D. Siraj *Mirror and Synchronous Couplings of Geometric Brownian Motions*

*Abstract: The paper studies the question of whether the classical mirror and synchronous couplings of two Brownian motions minimise and maximise, respectively, the coupling time of the corresponding geometric Brownian motions. We establish a characterisation of the optimality of the two couplings over any finite time horizon and show that, unlike in the case of Brownian motion, the optimality fails in general even if the geometric Brownian motions are martingales. On the other hand, we prove that in the cases of the ergodic average and the infinite time horizon criteria, the mirror coupling and the synchronous coupling are always optimal for general (possibly non-martingale) geometric Brownian motions. We show that the two couplings are efficient if and only if they are optimal over a finite time horizon and give a conjectural answer for the efficient couplings when they are suboptimal.*

To appear in Stochastic Processes & Applications.

*Keywords: mirror and synchronous coupling, coupling time, geometric Brownian motion, efficient coupling, optimal coupling, Bellman's principle.*

13-08**A. Jacquier, M. Lorig

*The Smile of certain Lévy-type Models**Abstract: We consider a class of assets whose risk-neutral pricing dynamics are described by an exponential L´evy-type process subject to default. The class of processes we consider features locally-dependent drift, diffusion and default-intensity as well as a locally-dependent L´evy measure. Using techniques from regular perturbation theory and Fourier analysis, we derive a series expansion for the price of a European-style option. We also provide precise conditions under which this series expansion converges to the exact price. *

*Additionally, for a certain subclass of assets in our modeling framework, we derive an expansion for the implied volatility induced by our option pricing formula. The implied volatility expansion is exact within its radius of convergence. As an example of our framework, we propose a class of CEV-like L´evy-type models. Within this class, approximate option prices can be computed by a single Fourier integral and approximate implied volatilities are explicit (i.e., no integration is required). Furthermore, the class of CEV-like L´evy-type models is shown to provide a tight fit to the implied volatility surface of S&P500 index options.*

*Keywords: Regular Perturbation, L´evy-type, Local Volatility, Implied Volatility, Default, CEV.*

13-09

A. Beskos, D. Crisan, A. Jasra, N. Whiteley**Error Bounds and Normalizing Constants for Sequential Monte Carlo Samplers in High Dimensions**

*Abstract: In this article we develop a collection of results associated to the analysis of the Sequential Monte Carlo (SMC) samplers algorithm, in the context of high-dimensional iid target probabilities. The SMC sampler algorithm can be designed to sample from a single probability distribution, using Monte Carlo to approximate expectations with respect to this law. Given a target density in d dimensions our results are concerned with d large while the number of Monte Carlo samples, N, remains fixed. We deduce an explicit bound on the Monte-Carlo error for estimates derived using the SMC sampler and the exact asymptotic relative L2-error of the estimate of the normalizing constant associated to the target. We also establish marginal propagation of chaos properties of the algorithm. These results are deduced when the cost of the algorithm is O(Nd^2).*

*Keywords: Sequential Monte Carlo, High Dimensions, Propagation of Chaos, Normalizing Constants.*

13-10

A. Beskos, D. Crisan, A. Jasra*On the Stability of Sequential Monte Carlo Methods in High Dimensions*

*Abstract: We investigate the stability of a Sequential Monte Carlo (SMC) method applied to the problem of sampling from a target distribution on Rd for large d. It is well known that using a single importance sampling step, one produces an approximation for the target that deteriorates as the dimension d increases, unless the number of Monte Carlo samples N increases at an exponential rate in d. We show that this degeneracy can be avoided by introducing a sequence of artificial targets, starting from a `simple' density and moving to the one of interest, using an SMC method to sample from the sequence. Using this class of SMC methods with a fixed number of samples, one can produce an approximation for which the effective sample size (ESS) converges to a random variable e_N as d increases. with 1 < e_N < N. The convergence is achieved with a computational cost proportional to Nd^2. If e_N<<N, we can raise its value by introducing a number of resampling steps, say m (where m is independent of d). Also, we show that the Monte Carlo error for estimating a fixed dimensional marginal expectation is of order 1/\sqrt{N} uniformly in d. The results imply that, in high dimensions, SMC algorithms can efficiently control the variability of the importance sampling weights and estimate fixed dimensional marginals at a cost which is less than exponential in d and indicate that resampling leads to a reduction in the Monte Carlo error and increase in the ESS. All of our analysis is made under the assumption that the target density is iid.*

*Keywords: Sequential Monte Carlo, High Dimensions, Resampling, Functional CLT.*

13-11

A. Jacquier, M. Lorig*From characteristic functions to implied volatility expansions*

*Abstract: For any strictly positive martingale S = eX for which X has an analytically tractable characteristic function, we provide an expansion for the implied volatility. This expansion is explicit in the sense that it involves no integrals, but only polynomials in log(K=S0). We illustrate the versatility of our expansion by computing the approximate implied volatility smile in three well-known martingale models: one nite activity exponential Levy model (Merton), one in nite activity exponential Levy model (Variance Gamma), and one stochastic volatility model (Heston). We show how this technique can be extended to compute approximate forward implied volatilities and we implement this extension in the Heston setting. Finally, we illustrate how our expansion can be used to perform a model-free calibration of the empirically observed implied volatility surface.*

*Keywords: Implied volatility expansions, exponential Levy, ane class, Heston, additive process.*

13-12

S. De Marco, C. Hillairet, A. Jacquier*Shapes of implied volatility with positive mass at zero*

*Abstract: We study the shapes of the implied volatility when the underlying distribution has an atom at zero. We show that the behaviour at small strikes is uniquely determined by the mass of the atom at least up to the third asymptotic order, regardless of the properties of the remaining (absolutely continuous, or singular) distribution on the positive real line. We investigate the structural difference with the no-mass-at-zero case, showing how one can-a priori-distinguish between mass at the origin and a heavy-left-tailed distribution. An atom at zero is found in stochastic models with absorption at the boundary, such as the CEV process, and can be used to model default events, as in the class of jump-to-default structural models of credit risk. We numerically test our model-free result in such examples. Note that while Lee's moment formula tells that implied variance is \emph{at most} asymptotically linear in log-strike, other celebrated results for exact smile asymptotics such as Benaim and Friz (09) or Gulisashvili (10) do not apply in this setting-essentially due to the breakdown of Put-Call symmetry-and we rely here on an alternative treatment of the problem.*

*Keywords: Atomic distribution, heavy-tailed distribution, Implied Volatility, smile asymptotics, absorption at zero, CEV model.*

### 2012

#### Working Papers 2012

12-01

S. Jacka, A. Mijatovic*Coupling and tracking of regime-switching martingales*

*Abstract: This paper describes two explicit couplings of standard Brownian motions $B$ and $V$, which naturally extend the mirror coupling and the synchronous coupling and respectively maximise and minimise (uniformly over all time horizons) the coupling time and the tracking error of two regime-switching martingales.*

*The generalised mirror coupling minimizes the coupling time of the two martingales while simultaneously maximising the tracking error for all time horizons. The generalised synchronous coupling maximises the coupling time and minimises the tracking error over all co-adapted couplings. The proofs are based on the Bellman principle.*

*We give counterexamples to the conjectured optimality of the two couplings amongst a wider classes of stochastic integrals.*

*Keywords: generalised mirror and synchronous coupling of Brownian motion, coupling time and tracking error of regime-switching martingales, Bellman principle, continuous-time Markov chains, stochastic integrals*

12-02

D. Crisan, O. Obanubi*Particle Filters with Random Resampling Times*

*Abstract: Particle filters are numerical methods for approximating the solution of the filtering problem which use systems of weighted particles that (typically) evolve according to the law of the signal process. These methods involve a corrective/resampling procedure which eliminates the particles that become redundant and multiplies the ones that contribute most to the resulting approximation. The correction is applied at instances in time called resampling/correction times. Practitioners normally use certain overall characteristics of the approximating system of particles (such as the effective sample size of the system) to determine when to correct the system. As a result, the resampling times are random. However, in the continuous time framework, all existing convergence results apply only to particle filters with deterministic correction times. In this paper, we analyse (continuous time) particle filters where resampling takes place at times that form a sequence of (predictable) stopping times. We prove that, under very general conditions imposed on the sequence of resampling times, the corresponding particle filters converge.*

*The conditions are verified when the resampling times are chosen in accordance to effective sample size of the system of particles, the coefficient of variation of the particles’ weights and, respectively, the (soft) maximum of the particles’ weights. We also deduce central-limit theorem type results for the approximating particle system with random resampling times.*

*Keywords: Stochastic partial differential equation, Filtering. Zakai equation, Particle filters, Sequential Monte-Carlo, Methods. Resampling, Resampling times, Random times, Effective Sample Size, Coefficient of variation, Soft Maximum, Central Limit Theorem.*

12-03

G. Pavliotis, A. Abdulle*Numerical Methods for Stochastic Partial Differential Equations with Multiple Scales*

*Abstract: A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations, Commun. Pure Appl. Math., 58(11) (2005) 1544–1585]. The class of problems that we consider are SPDEs with quadratic nonlinearities that were studied in [D. Blömker, M. Hairer, G.A. Pavliotis, Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities, Nonlinearity, 20(7) (2007) 1721–1744]. For such SPDEs an amplitude equation which describes the effective dynamics at long time scales can be rigorously derived for both advective and diffusive time scales. Our method, based on micro and macro solvers, allows to capture numerically the amplitude equation accurately at a cost independent of the small scales in the problem. Numerical experiments illustrate the behavior of the proposed method.*

*Keywords: Stochastic partial differential equations, Multiscale methods, Averaging, Homogenization, Heterogeneous multiscale method (HMM)*

12-04

M. Ottobre *Long time asymptotics of a Brownian Particle coupled with a random environment with non-diffusive feedback force*

(Stochastic Processes and their Applications, 122 (2012), 844-884)

*Abstract: We study the long time behavior of a Brownian particle moving in an anomalously diffusing field, the evolution of which depends on the particle position. We prove that the process describing the asymptotic behavior of the Brownian particle has bounded (in time) variance when the particle interacts with a subdiffusive field; when the interaction is with a superdiffusive field the variance of the limiting process grows in time as t2γ−1, 1/2 < γ < 1. Two different kinds of superdiffusing (random) environments are considered: one is described through the use of the fractional Laplacian; the other via the Riemann–Liouville fractional integral. The subdiffusive field is modeled through the Riemann–Liouville fractional derivative.*

*Keywords: Anomalous diffusion; Riemann–Liouville fractional derivative (integral); Fractional Laplacian; Continuous time random walk; Lévy flight; Scaling limit; Interface fluctuations.*

12-05

T. Cass, M. Hairer, C. Litterer, S. Tindel**Smoothness of the density for solutions to Gaussian rough differential equations (arXiv preprint)**

*Abstract: We consider stochastic differential equations of the form dYt = V (Yt) dXt+V0 (Yt) dt driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields V0 and V = (V1, . . . , Vd) satisfy H¨ormander’s bracket condition, we demonstrate that Yt admits a smooth density for any t 2 (0, T], provided the driving noise satisfies certain non-degeneracy assumptions. Our analysis relies on an interplay of rough path theory, Malliavin calculus, and the theory of Gaussian processes. Our result applies to a broad range of examples including fractional Brownian motion with Hurst parameter H > 1/4, the Ornstein-Uhlenbeck process and the Brownian bridge returning after time T.*

12-06

A. Mijatovic, M. Urusov *Convergence of Integral Functionals of One-Dimensional Diffusions*

Electronic Communications in Probability (to appear).

*Abstract: In this paper we describe the pathwise behaviour of the integral functional $\int_0^t f(Y_u)\,\dd u$ for any $t\in[0,\zeta]$, where $\zeta$ is (a possibly infinite) exit time of a one-dimensional diffusion process $Y$ from its state space, $f$ is a nonnegative Borel measurable function and the coefficients of the SDE solved by $Y$ are only required to satisfy weak local integrability conditions.*

*Two proofs of the deterministic characterisation of the convergence of such functionals are given: the problem is reduced in two different ways to certain path properties of Brownian motion where either the Williams theorem and the theory of Bessel processes or the first Ray-Knight theorem can be applied to prove the characterisation. As a simple application of the main results we give a short proof of Feller's test for explosion.*

*Keywords: Integral functional; one-dimensional diffusion; local time; Bessel process; Ray-Knight theorem; Williams theorem.*

12-07

*Markov chain approximations for transition densities of Levy processes*

*Abstract: We consider the convergence of a continuous-time Markov chain approximation $X^h$, $h>0$, to an $\mathbb{R}^d$-valued L'evy process $X$. The state space of $X^h$ is an equidistant lattice and its $Q$-matrix is chosen to approximate the generator of $X$. In dimension one ($d=1$), and then under a general sufficient condition for the existence of transition densities of $X$, we establish sharp convergence rates of the normalised probability mass function of $X^h$ to the probability density function of $X$. In higher dimensions ($d>1$), rates of convergence are obtained under a technical condition, which is satisfied when the diffusion matrix is non-degenerate.*

*Keywords: Levy process, continuous-time Markov chain, spectral representation, convergence rates for semi-groups and transition densities.*

12-08

F. Avram, A. Horvath, M. Pistorius *On matrix exponential approximations of the infimum of a spectrally negative Levy process*

*Abstract: We recall four open problems concerning constructing high-order matrix- exponential approximations for the in?mum of a spectrally negative Levy process (with applications to fi?rst-passage/ruin probabilities, the wait- ing time distribution in the M/G/1 queue, pricing of barrier options, etc).*

*On the way, we provide a new approximation, for the perturbed Cram?er- Lundberg model, and recall a remarkable family of (not minimal order) approximations of Johnson and Taa?e [JT89], which ?fit an arbitrarily high number of moments, greatly generalizing the currently used approximations of Renyi, De Vylder and Whitt-Ramsay. Obtaining such approximations which fit the Laplace transform at in?finity as well would be quite useful.*

**Keywords: **Levy process; ?first passage problem; Pollaczek-Khinchine formula; method of moments; matrix-exponential function; admissible Pad?e approximation; Johnson-Taaff?e approximations; two-point Pad?e approximations.

A. Jacquier, P. Roome*Asymptotics of forward implied volatility*

*Abstract: We prove here a general closed-form expansion formula for forward-start options and the forward implied volatility smile in a large class of models, including Heston and time-changed exponential Levy models. This expansion applies to both small and large maturities and is based solely on the knowledge of the forward characteristic function of the underlying process. The method is based on sharp large deviations techniques, and allows us to recover (in particular) many results for the spot implied volatility smile. In passing we show (i) that the small-maturity exploding behaviour of forward smiles depends on whether the quadratic variation of the underlying is bounded or not, and (ii) that the forward-start date also has to be rescaled in order to obtain non-trivial small-maturity asymptotics.*

*Keywords: implied volatility, forward volatility, forward-start, large deviations, saddlepoint methods.*

12-10

G. Guo, A. Jacquier, C. Martini, L. Neufcourt*Generalised arbitrage-free SVI volatility surfaces*

*Abstract: In this article we propose a generalisation of the recent work of Gatheral-Jacquier on explicit arbitrage-free parameterisations of implied volatility surfaces. We also discuss extensively the notion of arbitrage freeness and Roger Lee's moment formula using the recent analysis by Roper. We further exhibit an arbitrage-free volatility surface different from Gatheral's SVI parameterisation.*

*Keywords: implied volatility, no-arbitrage, SVI.*

12-11

J. Gatheral, A. Jacquier*Arbitrage-Free SVI Volatility Surfaces*

*Abstract: In this article, we show how to calibrate the widely-used SVI parameterization of the implied volatility smile in such a way as to guarantee the absence of static arbitrage. In particular, we exhibit a large class of arbitrage-free SVI volatility surfaces with a simple closed-form representation. We demonstrate the high quality of typical SVI fits with a numerical example using recent SPX options data.*

*Keywords: implied volatility, no-arbitrage, SVI.*

12-12

A. Jacquier, A. Mijatovic*Large deviations for the extended Heston model: the large-time case*

*Abstract: We study here the large-time behaviour of all continuous affine stochastic volatility models and deduce a closed-form formula for the large-maturity implied volatility smile. Based on refinements of the Gartner-Ellis theorem on the real line, our proof reveals pathological behaviours of the asymptotic smile. In particular, we show that the condition assumed in [10] under which the Heston implied volatility converges to the SVI parameterisation is necessary and sufficient.*

*Keywords: implied volatility, Heston model, asymptotics, large deviations.*

### 2011

**Working Papers 2011**

11-01

J.D. Deuschel, P.K. Friz, A. Jacquier, S. Violante*Marginal density expansions for diffusions and stochastic volatility*

*Abstract: Density expansions for hypoelliptic diffusions $(X^1,...,X^d)$ are revisited. In particular, we are interested in density expansions of the projection $(X_T^1,...,X_T^l)$, at time $T>0$, with $l \leq d$. Global conditions are found which replace the well-known "not-in-cutlocus" condition known from heat-kernel asymptotics; cf. G. Ben Arous (1988). Our small noise expansion allows for a "second order" exponential factor. Applications include tail and implied volatility asymptotics in some correlated stochastic volatility models; in particular, we solve a problem left open by A. Gulisashvili and E.M. Stein (2009).*

*Keywords: Laplace method onWiener space, generalized density expansions in small noise and small time, sub-Riemannian geometry with drift, focal points, stochastic volatility, implied volatility, large strike and small time asymptotics for implied volatility.*

11-02

M. Forde, A. Jacquier, A. Mijatovic*A note on essential smoothness in the Heston model*

Finance & Stochastics 15 (4): 781-784

*Abstract: This note studies an issue relating to essential smoothness that can arise when the theory of large deviations is applied to a certain option pricing formula in the Heston model. The note identifies a gap, based on this issue, in the proof of Corollary 2.4 in [2] and describes how to circumvent it. This completes the proof of Corollary 2.4 in [2] and hence of the main result in [2], which describes the limiting behaviour of the implied volatility smile in the Heston model far from maturity. *

11-03

A. Beskos, D. Crisan, A. Jasra, N. Whiteley**Error bounds and normalizing constants for sequential Monte Carlo in high dimensions**

*Abstract: In a recent paper [3], the Sequential Monte Carlo (SMC) sampler introduced in [12, 19, 24] has been shown to be asymptotically stable in the dimension of the state space d at a cost that is only polynomial in d, when N the number of Monte Carlo samples, is fixed. More precisely, it has been established that the effective sample size (ESS) of the ensuing (approximate) sample and the Monte Carlo error of fixed dimensional marginals will converge as d grows, with a computational cost of O(Nd2). In the present work, further results on SMC methods in high dimensions are provided as d ! 1 and with N fixed. We deduce an explicit bound on the Monte-Carlo error for estimates derived using the SMC sampler and the exact asymptotic relative L2-error of the estimate of the normalizing constant. We also establish marginal propagation of chaos properties of the algorithm. The accuracy in high-dimensions of some approximate SMC-based filtering schemes is also discussed.*

*Keywords: Sequential Monte Carlo, High Dimensions, Propagation of Chaos, Normalizing Constants, Filtering.*

11-04

G. Pavliotis, G. M. Pradas, D. Tseluiko, S. Kalliadasis, D. Papageorgiou* Noise induced state transitions, intermittency and universality in the noisy Kuramoto-Sivashinksy equation *Phys. Rev. Lett. 106, 060602

*Abstract: Consider the effect of pure additive noise on the long-time dynamics of the noisy Kuramoto- Sivashinsky (KS) equation close to the instability onset. When the noise acts only on the first stable mode (highly degenerate), the KS solution undergoes several state transitions, including critical on-off intermittency and stabilized states, as the noise strength increases. Similar results are obtained with the Burgers equation. Such noise-induced transitions are completely characterized through critical exponents, obtaining the same universality class for both equations, and rigorously explained using multiscale techniques.*

11-05

O. Barndorff-Nielsen, F. Benth, A. Veraart *Modelling Electricity Forward Markets by Ambit Fields*

*Abstract: This paper proposes a new modelling framework for electricity forward markets based on so-called ambit fields. The new model can capture many of the stylised facts observed in energy markets and is highly analytically tractable. We give a detailed account on the probabilistic properties of the new type of model, and we discuss martingale conditions, option pricing and change of measure within the new model class. Also, we derive a model for the typically stationary spot price, which is obtained from the forward model through a limiting argument.*

*Keywords: Electricity Markets, Forward Prices, Random Fields, Ambit Fields, Levy Basis, Samuelson Effect, Stochastic Volatility.*

11-06

O. Barndorff-Nielsen, A. Veraart **Stochastic Volatility of Volatility and Variance Risk Premia**

*Abstract: This paper introduces a new class of stochastic volatility models which allows for stochastic volatility of volatility (SVV): Volatility modulated non-Gaussian Ornstein-Uhlenbeck (VMOU) processes. Various probabilistic properties of (integrated) VMOU processes are presented. Further we study the effect of the SVV on the leverage effect and on the presence of long memory. One of the key results in the paper is that we can quantify the impact of the SVV on the (stochastic) dynamics of the variance risk premium (VRP). Moreover, provided the physical and the risk -- neutral probability measures are related through a structure -- preserving change of measure, we obtain an explicit formula for the VRP.*

*Keywords: Stochastic volatility of volatility, Levy process, Ornstein-Uhlenbeck process, variance risk premium, supOU process.*

11-07

T. Cass, C. Litterer, T. Lyons*Integrability estimates for Gaussian rough differential equations (arXiv preprint)*

*Abstract: We derive explicit tail-estimates for the Jacobian of the solution flow for stochastic differential equations driven by Gaussian rough paths. In particular, we deduce that the Jacobian has finite moments of all order for a wide class of Gaussian process including fractional Brownian motion with Hurst parameter H > 1/4. We remark on the relevance of such estimates to a number of significant open problems.*

### Previous Papers

10-01

Crisan D, Manolarakis K *Second order discretization of Backward SDEs*

*Abstract: In [5] the authors suggested a new algorithm for the numerical approximation of a BSDE by merging the cubature method with the first order discretization developed by [3] and [16]. Though the algorithm presented in [5] compared satisfactorily with other methods it lacked the higher order nature of the cubature method due to the use of the low order discretization. In this paper we introduce a second order discretization of the BSDE in the spirit of higher order implicit-explicit schemes for forward SDEs and predictor corrector methods. <\p>*

*Keywords: Backward SDEs, Second order discretization, Numerical analysis.*

07-01

Pavliotis G, Stuart A.M.**Parameter Estimation for Multiscale Diffusions. J. Stat. Phys. 127(4) 741-781**

*Abstract: We study the problem of parameter estimation for time-series possessing two, widely separated, characteristic time scales. The aim is to understand situations where it is desirable to fit a homogenized single-scale model to suchmultiscale data.We demonstrate, numerically and analytically, that if the data is sampled too finely then the parameter fit will fail, in that the correct parameters in the homogenized model are not identified.We also show, numerically and analytically, that if the data is subsampled at an appropriate rate then it is possible to estimate the coefficients of the homogenized model correctly. The ideas are studied in the context of thermally activated motion in a two-scale potential. However the ideas may be expected to transfer to other situations where it is desirable to fit an averaged or homogenized equation to multiscale data.*

*Keywords: Parameter estimation, multiscale diffusions, stochastic differentialequations, homogenization, maximum likelihood, subsampling.*

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