# 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

### 2018

**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.

### 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).

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### 2016

16-01

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.

16-04

John Armstrong, Damiano Brigo

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

We explain how Itô Stochastic Differential Equations on manifolds may be defined 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 different types of drawings. We explain that the mainstream choice of Fisk-Stratonovich-McShane calculus for stochastic differential geometry is not necessary and elaborate on the relationships with the jets approach. We consider the two calculi as being simply different coordinate systems for the same underlying coordinate-free stochastic differential equation. If the extrinsic approach to differential 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 dictated by the type of problem one is facing and the related desiderata. We also discuss the forward Kolmogorov equation and the backward diffusion operator in geometric terms, and consider percentiles of the solutions of the SDE and their properties, leading to fan diagrams and their relationship with jets. In particular, the median of an 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.

16-05

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].

16-06

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 finite dimensional and infinite dimensional terminology for exponential and mixture spaces are related.

16-07

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.

16-08**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.

16-09**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.

16-10**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.

16-11**Joint Asymptotic Distribution of Certain Path Functionals of the Reflected Process**Aleksandar Mijatović, Martijn Pistorius

See link '16-11' above for the abstract and paper.

16-12**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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