## Publications

18 results found

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

Given a Gaussian process $X$, its canonical geometric rough path lift$\mathbf{X}$, and a solution $Y$ to the rough differential equation (RDE)$\mathrm{d}Y_{t} = V\left (Y_{t}\right ) \circ \mathrm{d} \mathbf{X}_t$, wepresent a closed-form correction formula for $\int Y \circ \mathrm{d}\mathbf{X} - \int Y \, \mathrm{d} X$, i.e. the difference between the rough andSkorohod integrals of $Y$ with respect to $X$. When $X$ is standard Brownianmotion, 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 familiarexamples, including fractional Brownian motion with $H > \frac{1}{3}$. To provethe formula, we first show that the Riemann-sum approximants of the Skorohodintegral converge in $L^2(\Omega)$ by using a novel characterization of theCameron-Martin norm in terms of higher-dimensional Young-Stieltjes integrals.Next, we append the approximants of the Skorohod integral with a suitablecompensation term without altering the limit, and the formula is finallyobtained after a re-balancing of terms.

Cass T, Weidner MP, Hörmander's theorem for rough differential equations on manifolds

We introduce a new definition for solutions $Y$ to rough differentialequations (RDEs) of the form $\mathrm d Y_{t} =V\left (Y_{t}\right )\mathrmd\mathbf X_{t} ,Y_{0} =y_{0}\text{}$. By using the Grossman-Larson Hopf algebraon labelled rooted trees, we prove equivalence with the classical definition ofa solution advanced by Davie when the state space $E$ for $Y$ is afinite-dimensional vector space. The notion of solution we propose, however,works when $E$ is any smooth manifold $\mathcal{M}$ and is therefore equallywell-suited for use as an intrinsic defintion of an $\mathcal{M}$-valued RDEsolution. This enables us to prove existence, uniqueness andcoordinate-invariant theorems for RDEs on $\mathcal{M}$ bypassing the need todefine a rough path on $\mathcal{M}$. Using this framework, we generaliseresult of Cass, Hairer, Litter and Tindel proving the smoothness of the densityof $\mathcal{M}$-valued RDEs driven by non-degenerate Gaussian rough pathsunder H\"ormander's bracket condition. In doing so, we reinterpret some of thefoundational results of the Malliavin calculus to make them appropriate to thestudy $\mathcal{M}$-valued Wiener functionals.

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

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- Citations: 1

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, Hairer M, Litterer C,
et al., 2015, SMOOTHNESS OF THE DENSITY FOR SOLUTIONS TO GAUSSIAN ROUGH DIFFERENTIAL EQUATIONS, *ANNALS OF PROBABILITY*, Vol: 43, Pages: 188-239, ISSN: 0091-1798

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- Citations: 16

Cass T, Lyons T, 2015, Evolving communities with individual preferences, *PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY*, Vol: 110, Pages: 83-107, ISSN: 0024-6115

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- Citations: 3

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

© Springer International Publishing Switzerland 2014. 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, 2014, ON THE ERROR ESTIMATE FOR CUBATURE ON WIENER SPACE, *PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY*, Vol: 57, Pages: 377-391, ISSN: 0013-0915

Cass T, Litterer C, Lyons T, 2013, INTEGRABILITY AND TAIL ESTIMATES FOR GAUSSIAN ROUGH DIFFERENTIAL EQUATIONS, *ANNALS OF PROBABILITY*, Vol: 41, Pages: 3026-3050, ISSN: 0091-1798

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- Citations: 17

Cass T, Litterer C, Lyons T, 2012, Rough Paths on Manifolds

We develop a fundamental framework for and extend the theory of rough pathsto Lipschitz-gamma manifolds.

Cass T, Friz P, 2011, Malliavin calculus and rough paths, *BULLETIN DES SCIENCES MATHEMATIQUES*, Vol: 135, Pages: 542-556, ISSN: 0007-4497

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- Citations: 1

Cass T, Friz P, 2010, Densities for rough differential equations under Hormander's condition, *ANNALS OF MATHEMATICS*, Vol: 171, Pages: 2115-2141, ISSN: 0003-486X

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- Citations: 31

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

We prove existence and uniqueness results for (mild) solutions to somenon-linear parabolic evolution equations with a rough forcing term. Our methodof proof relies on a careful exploitation of the interplay between the spatialand time regularity of the solution by capitialising some of Kato's ideas insemigroup theory. Classical Young integration theory is then shown to provide ameans of interpreting the equation. As an application we consider the threedimensional Navier-Stokes system with a stochastic forcing term arising from afractional Brownian motion with h > 1/2.

Cass T, 2009, Smooth densities for solutions to stochastic differential equations with jumps, *STOCHASTIC PROCESSES AND THEIR APPLICATIONS*, Vol: 119, Pages: 1416-1435, ISSN: 0304-4149

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- Citations: 12

Cass T, Friz P, Victoir N, 2009, NON-DEGENERACY OF WIENER FUNCTIONALS ARISING FROM ROUGH DIFFERENTIAL EQUATIONS, *TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY*, Vol: 361, Pages: 3359-3371, ISSN: 0002-9947

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- Citations: 24

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

We extend the Bismut-Elworthy-Li formula to non-degenerate jump diffusionsand "payoff" functions depending on the process at multiple future times. Inthe spirit of Fournie et al [13] and Davis and Johansson [9] this can improveMonte Carlo numerics for stochastic volatility models with jumps. To this endone needs so-called Malliavin weights and we give explicit formulae valid inpresence of jumps: (a) In a non-degenerate situation, the extended BEL formularepresents possible Malliavin weights as Ito integrals with explicitintegrands; (b) in a hypoelliptic setting we review work of Arnaudon andThalmaier [1] and also find explicit weights, now involving the Malliavincovariance matrix, but still straight-forward to implement. (This is incontrast to recent work by Forster, Lutkebohmert and Teichmann where weightsare constructed as anticipating Skorohod integrals.) We give some financialexamples covered by (b) but note that most practical cases of poor Monte Carloperformance, Digital Cliquet contracts for instance, can be dealt with by theextended BEL formula and hence without any reliance on Malliavin calculus atall. We then discuss some of the approximations, often ignored in theliterature, needed to justify the use of the Malliavin weights in the contextof standard jump diffusion models. Finally, as all this is meant to improvenumerics, we give some numerical results with focus on Cliquets under theHeston model with jumps.

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