## Publications

30 results found

Salvi C, Cass T, Foster J,
et al., 2021, The signature kernel is the solution of a Goursat PDE, *SIAM Journal on Mathematics of Data Science*, ISSN: 2577-0187

Recently, there has been an increased interest in the development of kernel methods for learning with sequential data. The signature kernel is a learning tool with potential to handle irregularly sampled, multivariate time series. In [1] the authors introduced a kernel trick for the truncated version of this kernel avoiding the exponential complexity that would have been involved in a direct computation. Here we show that for continuously differentiable paths, the signature kernel solves a hyperbolic PDE and recognize the connection with a well known class of differential equations known in the literature as Goursat problems. This Goursat PDE only depends on the increments of the input sequences, does not require the explicit computation of signatures and can be solved efficiently using state-of-the-art hyperbolic PDE numerical solvers, giving a kernel trick for the untruncated signature kernel, with the same raw complexity as the method from [1], but with the advantage that the PDE numerical scheme is well suited for GPU parallelization, which effectively reduces the complexity by a full order of magnitude in the length of the input sequences. In addition, we extend the previous analysis to the space of geometric rough paths and establish, using classical results from rough path theory, that the rough version of the signature kernel solves a rough integral equation analogous to the aforementioned Goursat problem. Finally, we empirically demonstrate the effectiveness of this PDE kernel as a machine learning tool in various data science applications dealing with sequential data. We release the library sigkernel publicly available at https://github.com/crispitagorico/sigkernel

Lemercier M, Salvi C, Cass T, et al., 2021, SigGPDE: scaling sparse Gaussian processes on sequential data, Thirty-eighth International Conference on Machine Learning (ICML-2021), ISSN: 2640-3498

Making predictions and quantifying their uncertainty when the input data is sequential is a fundamental learning challenge, recently attracting increasing attention. We develop SigGPDE, a new scalable sparse variational inference framework for Gaussian Processes (GPs) on sequential data. Our contribution is twofold. First, we construct inducing variables underpinning the sparse approximation so that the resulting evidence lower bound (ELBO) does not require any matrix inversion. Second, we show that the gradients of the GP signature kernel are solutions of a hyperbolic partial differential equation (PDE). This theoretical insight allows us to build an efficient back-propagation algorithm to optimize the ELBO. We showcase the significant computational gains of SigGPDE compared to existing methods, while achieving state-of-the-art performance for classification tasks on large datasets of up to1millionmultivariate time series.

Armstrong J, Bellani C, Brigo D,
et al., 2021, Option pricing models without probability: a rough paths approach, *Mathematical Finance*, ISSN: 0960-1627

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

Armstrong J, Bellani C, Brigo D, et al., 2021, Option pricing models without probability: A rough paths approach, Publisher: WILEY

Cass T, Lim N, 2021, Skorohod and rough integration for stochastic differential equations driven by Volterra processes, *L'Institut Henri Poincare, Annales B: Probabilites et Statistiques*, Vol: 57, Pages: 132-168, ISSN: 0246-0203

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

Cass T, Crisan D, Dobson P, et al., 2021, Long-time behaviour of degenerate diffusions: UFG-type SDEs and time-inhomogeneous hypoelliptic processes, Publisher: INST MATHEMATICAL STATISTICS-IMS

Cass T, Crisan D, Ottobre M,
et al., 2020, Long-time behaviour of degenerate diffusions: UFG-type SDEs and time-inhomogeneous hypoelliptic processes, *Electronic Journal of Probability*, ISSN: 1083-6489

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

Bensoussan A, Cass T, Chau MHM,
et al., 2020, Mean field games with parametrized followers, *IEEE Transactions on Automatic Control*, Vol: 65, Pages: 12-27, ISSN: 0018-9286

In this article, we consider mean field games between a dominant leader and a large group of followers, such that each follower is subject to a heterogeneous delay effect from the action of the leader, who in turn can exercise governance on the population through this influence. We assume that the delay effects are discretely distributed among the followers. Given regular enough coefficients, we describe a necessary condition for the existence of a solution for the equilibrium by a system of coupled forward-backward stochastic differential equations and stochastic partial differential equations. We provide a thorough study for the particular Linear Quadratic case. By adopting a functional approach, we obtain the time-independent sufficient condition which warrants the unique existence of the solution of the whole mean field game problem. Several numerical illustrations with different time horizons and populations are demonstrated.

Cass T, Lim N, 2019, A Stratonovich-Skorohod integral formula for Gaussian rough paths, *Annals of Probability*, Vol: 47, Pages: 1-60, ISSN: 0091-1798

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, that is, 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-Itô 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 L2(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 rebalancing of terms.

Cass T, Ogrodnik M, 2017, Tail estimates for Markovian rough paths, *Annals of Probability*, Vol: 45, Pages: 2477-2504, ISSN: 0091-1798

The accumulated local p-variation functional [Ann. Probab. 41 (213) 3026–3050] arises naturally in the theory of rough paths in estimates both for solutions to rough differential equations (RDEs), and for the higher-order terms of the signature (or Lyons lift). In stochastic examples, it has been observed that the tails of the accumulated local p-variation functional typically decay much faster than the tails of classical p-variation. This observation has been decisive, for example, for problems involving Malliavin calculus for Gaussian rough paths [Ann. Probab. 43 (2015) 188–239].All of the examples treated so far have been in this Gaussian setting that contains a great deal of additional structure. In this paper, we work in the context of Markov processes on a locally compact Polish space E, which are associated to a class of Dirichlet forms. In this general framework, we first prove a better-than-exponential tail estimate for the accumulated local p-variation functional derived from the intrinsic metric of this Dirichlet form. By then specialising to a class of Dirichlet forms on the step ⌊p⌋ free nilpotent group, which are sub-elliptic in the sense of Fefferman–Phong, we derive a better than exponential tail estimate for a class of Markovian rough paths. This class includes the examples studied in [Probab. Theory Related Fields 142 (2008) 475–523]. We comment on the significance of these estimates to recent papers, including the results of Ni Hao [Personal communication (2014)] and Chevyrev and Lyons [Ann. Probab. To appear].

Cass T, Driver BK, Lim N,
et al., 2016, On the integration of weakly geometric rough paths, *Journal of the Mathematical Society of Japan*, Vol: 68, Pages: 1505-1524, ISSN: 0025-5645

We close a gap in the theory of integration for weakly ge-ometric rough paths in the infinite-dimensional setting. We show that theintegral of a weakly geometric rough path against a sufficiently regular one form is, once again, a weakly geometric rough path.

Cass T, Driver BK, Litterer C, 2015, Constrained Rough Paths, *Proceedings of the London Mathematical Society*, Vol: 111, Pages: 1471-1518, ISSN: 1460-244X

We introduce a notion of rough paths on embedded submanifolds and demonstratethat this class of rough paths is natural. On the way we develop a notion ofrough integration and an efficient and intrinsic theory of rough differentialequations (RDEs) on manifolds. The theory of RDEs is then used to constructparallel translation along manifold valued rough paths. Finally, this frameworkis used to show there is a one to one correspondence between rough paths on ad-dimensional manifold and rough paths on d-dimensional Euclidean space. Thislast result is a rough path analogue of Cartan's development map and itsstochastic version which was developed by Eeels and Elworthy and Malliavin.

Cass T, Lyons T, 2015, Evolving communities with individual preferences, *Proceedings of the London Mathematical Society*, Vol: 110, Pages: 83-107, ISSN: 0024-6115

The goal of this paper is to provide mathematically rigorous tools for modelling the evolution of a community of interacting individuals. We model the population by a measure space (𝛺,,𝜈) where 𝜈 determines the abundance of individual preferences. The preferences of an individual 𝜔∈𝛺 are described by a measurable choice 𝑋(𝜔)of a rough path.We aim to identify, for each individual, a choice for the forward evolution 𝑌𝑡(𝜔)for an individual in the community. These choices 𝑌𝑡(𝜔) must be consistent so that 𝑌𝑡(𝜔)correctly accounts for the individual's preference and correctly models their interaction with the aggregate behaviour of the community.In general, solutions are continuum of interacting threads analogous to the huge number of individual atomic trajectories that together make up the motion of a fluid. The evolution of the population need not be governed by any over‐arching partial differential equation (PDE). Although one can match the standard non‐linear parabolic PDEs of McKean–Vlasov type with specific examples of communities in this case. The bulk behaviour of the evolving population provides a solution to the PDE.We focus on the case of weakly interacting systems, where we are able to exhibit the existence and uniqueness of consistent solutions.An important technical result is continuity of the behaviour of the system with respect to changes in the measure 𝜈assigning weight to individuals. Replacing the deterministic 𝜈 with the empirical distribution of an independent and identically distributed sample from 𝜈leads to many standard models, and applying the continuity result allows easy proofs for propagation of chaos.The rigorous underpinning presented here leads to uncomplicated models which have wide applicability in both the physical and social sciences. We make no presumption that the macroscopic dynamics are modelled by a PDE.This work builds on the fine probability literature considering the limit behaviour for systems where

Cass T, Clark M, Crisan D, 2014, The filtering equations revisited, *Springer Proceedings in Mathematics and Statistics*, Vol: 100, Pages: 129-162, ISSN: 2194-1009

The problem of nonlinear filtering has engendered a surprising number of mathematical techniques for its treatment. A notable example is the changeof– probability-measure method introduced by Kallianpur and Striebel to derive the filtering equations and the Bayes-like formula that bears their names. More recent work, however, has generally preferred other methods. In this paper, we reconsider the change-of-measure approach to the derivation of the filtering equations and show that many of the technical conditions present in previous work can be relaxed. The filtering equations are established for general Markov signal processes that can be described by amartingale-problem formulation.Two specific applications are treated.

Cass T, Litterer C, Lyons T, 2013, Integrability and tail estimates for Gaussian rough differential equations, *Annals of Probability*, Vol: 41, Pages: 3026-3050

Cass T, Litterer C, Lyons T, 2012, Rough Paths on Manifolds, New trends in stochastic analysis and related topics, 33–88,, Editors: Zhao, Truman, Publisher: World Scientific Publishing Company, Pages: 33-88, ISBN: 9789814360913

The volume is dedicated to Professor David Elworthy to celebrate his fundamental contribution and exceptional influence on stochastic analysis and related fields.

Cass T, Hairer M, Litterer C, et al., 2012, Smoothness of the density for solutions to Gaussian Rough Differential Equations

Cass T, Friz P, 2011, Malliavin calculus and rough paths, *Bulletin des Sciences Mathematiques*, Vol: 6-7, Pages: 542-556

Cass T, Friz P, 2010, Densities for rough differential equations under Hörmander’s condition, *Annals of Mathematics*, Vol: 171, Pages: 2115-2141, ISSN: 0003-486X

Cass T, Qian Z, Tudor J, 2010, Non-Linear Evolution Equations Driven by Rough Paths

Cass T, 2009, Smooth densities for solutions to stochastic differential equations with jumps, *Stochastic Processes and their Applications*, Vol: 119

Cass T, Friz P, Victoir N, 2009, Non-degeneracy of Wiener functionals arising from rough differential equations, *Transactions of the American Mathematical Society*, Pages: 3359-3371

Cass T, Friz P, 2007, The Bismut-Elworthy-Li formula for jump-diffusions and applications to Monte Carlo pricing in finance

Sabir I, Fraser J, Cass T,
et al., 2007, A quantitative analysis of the effect of cycle length on arrthymogenicity in hypokaleamic Langendorff-perfused murine hearts, *Pflugers Archiv-European Journal of Physiology*, Vol: 6, Pages: 925-936

Armstrong J, Brigo D, Cass T, et al., Non-Geometric Rough Paths on Manifolds

We provide a theory of manifold-valued rough paths of bounded 3 >p-variation, which we do not assume to be geometric. Rough paths are defined incharts, and coordinate-free (but connection-dependent) definitions of the roughintegral of cotangent bundle-valued controlled paths, and of RDEs driven by arough path valued in another manifold, are given. When the path is therealisation of semimartingale we recover the theory of It\^o integration andSDEs on manifolds [\'E89]. We proceed to present the extrinsic counterparts toour local formulae, and show how these extend the work in [CDL15] to thesetting of non-geometric rough paths and controlled integrands more generalthan 1-forms. In the last section we turn to parallel transport and Cartandevelopment: the lack of geometricity leads us to make the choice of aconnection on the tangent bundle of the manifold TM, which figures in an It\^ocorrection term in the parallelism RDE; such connection, which is not needed inthe geometric/Stratonovich setting, is required to satisfy properties whichguarantee well-definedness, linearity, and optionally isometricity of paralleltransport. We conclude by providing numerous examples, some accompanied bynumerical simulations, which explore the additional subtleties introduced byour change in perspective.

Cass T, Lim N, A Stratonovich-Skorohod integral formula for Volterra Gaussian rough paths

Given a solution $Y$ to a rough differential equation (RDE), a recent result[8] extends the classical It\"{o}-Stratonovich formula and provides aclosed-form expression for $\int Y \circ \mathrm{d} \mathbf{X} - \int Y \,\mathrm{d} X$, 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 Gaussianprocesses with finite $p$-variation such that $3 \leq p < 4$. The constraintthis time is that we restrict ourselves to Volterra Gaussian processes withkernels satisfying a natural condition, which however still allows the resultto encompass many standard examples, including fractional Brownian motion with$H > \frac{1}{4}$. Analogously to [8], we first show that the Riemann-sumapproximants of the Skorohod integral converge in $L^2(\Omega)$ by adopting asuitable characterization of the Cameron-Martin norm, before appending theapproximants with higher-level compensation terms without altering the limit.Lastly, the formula is obtained after a re-balancing of terms, and we also showhow to recover the standard It\"{o} formulas in the case where the vectorfields of the RDE governing $Y$ are commutative.

Cass T, Weidner MP, Tree algebras over topological vector spaces in rough path theory

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

Cass T, Reis GD, Salkeld W, Rough functional quantization and the support of McKean-Vlasov equations

We prove a representation for the support of McKean Vlasov Equations. To doso, we construct functional quantizations for the law of Brownian motion as ameasure over the (non-reflexive) Banach space of H\"older continuous paths. Bysolving optimal Karhunen Lo\`eve expansions and exploiting the compactembedding of Gaussian measures, we obtain a sequence of deterministic finitesupported measures that converge to the law of a Brownian motion with explicitrate. We show the approximation sequence is near optimal with very favourableintegrability properties and prove these approximations remain true when thepaths are enhanced to rough paths. These results are of independent interest. The functional quantization results then yield a novel way to builddeterministic, finite supported measures that approximate the law of the McKeanVlasov Equation driven by the Brownian motion which crucially avoid the use ofrandom empirical distributions. These are then used to solve an approximateskeleton process that characterises the support of the McKean Vlasov Equation. We give explicit rates of convergence for the deterministic finite supportedmeasures in rough-path H\"older metrics and determine the size of the particlesystem required to accurately estimate the law of McKean Vlasov equations withrespect to the H\"older norm.

Salvi C, Cass T, Foster J, et al., The Signature Kernel is the solution of a Goursat PDE

Recently, there has been an increased interest in the development of kernelmethods for learning with sequential data. The signature kernel is a learningtool with potential to handle irregularly sampled, multivariate time series. In"Kernels for sequentially ordered data" the authors introduced a kernel trickfor the truncated version of this kernel avoiding the exponential complexitythat would have been involved in a direct computation. Here we show that forcontinuously differentiable paths, the signature kernel solves a hyperbolic PDEand recognize the connection with a well known class of differential equationsknown in the literature as Goursat problems. This Goursat PDE only depends onthe increments of the input sequences, does not require the explicitcomputation of signatures and can be solved efficiently usingstate-of-the-arthyperbolic PDE numerical solvers, giving a kernel trick for theuntruncated signature kernel, with the same raw complexity as the method from"Kernels for sequentially ordered data", but with the advantage that the PDEnumerical scheme is well suited for GPU parallelization, which effectivelyreduces the complexity by a full order of magnitude in the length of the inputsequences. In addition, we extend the previous analysis to the space ofgeometric rough paths and establish, using classical results from rough paththeory, that the rough version of the signature kernel solves a rough integralequation analogous to the aforementioned Goursat PDE. Finally, we empiricallydemonstrate the effectiveness of our PDE kernel as a machine learning tool invarious machine learning applications dealing with sequential data. We releasethe library sigkernel publicly available athttps://github.com/crispitagorico/sigkernel.

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