Thomas Cass

NON-DEGENERACY OF WIENER FUNCTIONALS ARISING FROM ROUGH DIFFERENTIAL EQUATIONS

Malliavin Calculus is about Sobolev-type regularity of functionals on Wiener space, the main example being the Ito map obtained by solving stochastic differential equations. Rough path analysis is about strong regularity of the solution to (possibly stochastic) differential equations. We combine arguments of both theories and discuss the existence of a density for solutions to stochastic differential equations driven by a general class of non-degenerate Gaussian processes, including processes with sample path regularity worse than Brownian motion.

Smoothness of the density for solutions to Gaussian rough differential equations

We consider stochastic differential equations driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields satisfy Hormanders bracket condition, we demonstrate that the solution admits a smooth density for any strictly positive time 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 greater than 1/4, the Ornstein-Uhlenbeck process and the Brownian bridge returning after time T.

INTEGRABILITY AND TAIL ESTIMATES FOR GAUSSIAN ROUGH DIFFERENTIAL EQUATIONS

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.

Evolving communities with individual preferences

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, F ,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 )m ust be consistent so thatYt(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 ν assigning weight to individuals. Replacing the deterministic ν with the empirical distribution of an independent and identically distributed sample from ν leads 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 a large number of particles are interacting with independent preferences; there is also work on continuum models with preferences described by a semi-martingale measure. We mention some of the key papers.

Constrained Rough Paths

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 dierential 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.

On The Error Estimate for Cubature on Wiener Space

It was pointed out in Crisan, Ghazali [2] that the error estimate for the cubature on Wiener space algorithm developed in Lyons, Victoir [11] requires an additional assumption on the drift. In this note we demonstrate that it is straightforward to adopt the analysis of Kusuoka [7] to obtain a general estimate without an additional assumptions on the drift. In the process we slightly sharpen the bounds derived in [7].

Tail estimates for Markovian rough paths

We work in the context of Markovian rough paths associated to a class of uniformly subelliptic Dirichlet forms ([26]) and prove a better-than-exponential tail estimate for the accumulated local p-variation functional, which has been introduced and studied in [17]. We comment on the significance of these estimates to a range of currently-studied problems, including the recent results of Ni Hao [32], and Chevyrev and Lyons [18].

The Filtering Equations Revisited

The problem of nonlinear filtering has engendered a surprising number of mathematical techniques for its treatment. A notable example is the change-of–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 a martingale-problem formulation. Two specific applications are treated.