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We consider controlled ordinary differential equations and give new estimates for higher order Euler schemes. Our proofs are inspired by recent work of A.M. Davie who considers first and second order schemes. In order to implement the general case we make systematic use of geodesic approximations in the free nilpotent group. Such Euler estimates have powerful applications. By a simple limit argument they apply to rough path differential equations (RDEs) in the sense of T. Lyons and hence also to stochastic differential equations driven by Brownian motion or other random rough paths with sufficient integrability. In the context of the latter, we obtain strong remainder estimates in stochastic Taylor expansions a la Azencott, Ben Arous, Castell and Platen. Although our findings appear novel even in the case of driving Brownian motion our main insight is the genuine rough path nature of (quantitative) remainder estimates in stochastic Taylor expansions. There are several other applications of which we discuss in detail L q -convergence in Lyons' Universal Limit Theorem and moment control of RDE solutions.

Let X a be a Markov process with generator ∑i,j∂i( aij∂j· ) where a is a uniformly elliptic symmetric matrix. Thanks to the fundamental works of T. Lyons, stochastic differential equations driven by X a can be solved in the “rough path sense”; that is, pathwise by using a suitable stochastic area process. Our construction of the area, which generalizes previous works of Lyons–Stoica and then Lejay, is based on Dirichlet forms associated to subellitpic operators. This enables us in particular to discuss large deviations and support descriptions in suitable rough path topologies. As typical rough path corollary, Freidlin–Wentzell theory and the Stroock–Varadhan support theorem remain valid for stochastic differential equations driven by X a.

We study large deviation principles for Gaussian processes lifted to the free nilpotent group of step N. We apply this to a large class of Gaussian processes lifted to geometric rough paths. A large deviation principle for enhanced (fractional) Brownian motion, in Holder- or modulus topology, appears as special case.

For a d-dimensional diffusion of the form dXt = �( Xt)dt + σ(Xt)dWt and continuous functions f and g, we study the existence and uniqueness of adapted processes Y, Z, Ŵ, and A solving the second-order backward stochastic differential equation (2BSDE) dYt = f (t, Xt,Yt, Zt, Ŵt)dt + Z ′ ◦ dXt, t ∈ [0,T ), dZt = At dt + Ŵt dXt

We use simple sub-Riemannian techniques to prove that every weak geometric p-rough path (a geometric p-rough path in the sense of [20]) is the limit in sup-norm of a sequence of canonically lifted smooth paths, uniformly bounded in p-variation, thus clarifying the two different definitions of a geometric p-rough path. Our proofs are sufficiently general to include the case of Holder- and modulus-type regularity. This allows us to extend a few classical results on Holder-spaces and p -variation spaces to the non-commutative setting necessary for the theory of rough paths. As an application, we give a precise description of the support of Enhanced Fractional Brownian Motion, and prove a conjecture by Ledoux et al.

A geometric p-rough path can be seen to be a genuine path of finite p-variation with values in a Lie group equipped with a natural distance. The group and its distance lift ( R d , + , 0 ) and its Euclidean distance. This approach allows us to easily get a precise modulus of continuity for the Enhanced Brownian Motion (the Brownian Motion and its Levy Area). As a first application, extending an idea due to Millet and Sanz-Sole, we characterize the support of the Enhanced Brownian Motion (without relying on correlation inequalities). Secondly, we prove Schilder's theorem for this Enhanced Brownian Motion. As all results apply in Holder (and stronger) topologies, this extends recent work by Ledoux, Qian, Zhang [Stochastic Process. Appl. 102 (2) (2002) 265–283]. Lyons' fine estimates in terms of control functions [Rev. Mat. Iberoamericana 14 (2) (1998) 215–310] allow us to show that the Ito map is still continuous in the topologies we introduced. This provides new and simplified proofs of the Stroock–Varadhan support theorem and the Freidlin–Wentzell theory. It also provides a short proof of modulus of continuity for diffusion processes along old results by Baldi.

This note extends the work of Capitaine (J. Funct. Anal. 179 (1) (2001) 153) on the Levy area process for the free Brownian motion in two directions. First, we reprove that a Levy area for the Free Brownian motion exists in the Von Neumann tensor product, by exhibiting a non-commutative Burkholder–Davis–Gundy type inequality. Then, we show that there does not exist a Levy area in the projective tensor product.

We discuss $p$-variation regularity of real-valued functions defined on $[0,T]\times [0,T]$, based on rectangular increments. When $p \gt 1$, there are two slightly different notions of $p$-variation; both of which are useful in the context of Gaussian roug paths. Unfortunately, these concepts were blurred in previous works; the purpose of this note is to show that the afore-mentioned notions of $p$-variations are "epsilon-close". In particular, all arguments relevant for Gaussian rough paths go through with minor notational changes.

Nous donnons une condition simple et optimale sur la covariance d’un processus gaussien pour que celui-ci puisse etre associe naturellement a un rough path. Une fois ce processus construit, nous demontrons un principe de grandes deviations, un theoreme du support, et plusieurs resultats d’approximations. Avec la theorie des rough paths de T. Lyons, nous obtenons ainsi un cadre puissant, bien que conceptuellement simple, dans lequel nous pouvons analyser les equations differentielles conduites par des signaux gaussiens dans le sens des rough paths.

International audience; We propose some construction of enhanced Gaussian processes using Karhunen-Loeve expansion. We obtain a characterization and some criterion of existence and uniqueness. Using rough-path theory, we derive some Wong-Zakai Theorem.

We show that any continuous path of finite p-variation can be lifted to a geometric q-rough path, where q > p .

We consider anticipative Stratonovich stochastic differential equations driven by some stochastic process lifted to a rough path. Neither adaptedness of initial point and vector fields nor commuting conditions between vector field is assumed. Under a simple condition on the stochastic process, we show that the unique solution of the above SDE understood in the rough path sense is actually a Stratonovich solution. We then show that this condition is satisfied by the Brownian motion. As application, we obtain rather flexible results such as support theorems, large deviation principles and Wong--Zakai approximations for SDEs driven by Brownian motion along anticipating vectorfields. In particular, this unifies many results on anticipative SDEs., Published at http://dx.doi.org/10.1214/009117906000000827 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Malliavin Calculus is about Sobolev-type regularity of functionals on Wiener space, the main example being the Itô 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.

The authors present a new simple algorithm to approximate weakly stochastic differential equations in the spirit of [1] and [2]. They apply it to the problem of pricing Asian options under the Heston stochastic volatility model, and compare it with other known methods. It is shown that the combination of the suggested algorithm and quasi-Monte Carlo methods makes computations extremely fast. [1] Shigeo Kusuoka, ``Approximation of Expectation of Diffusion Process and Mathematical Finance,'' Advanced Studies in Pure Mathematics, Proceedings of Final Taniguchi Symposium, Nara 1998 (T. Sunada, ed.), vol. 31 2001, pp. 147--165. [2] Terry Lyons and Nicolas Victoir, ``Cubature on Wiener Space,'' Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 460 (2004), pp. 169--198., 15 pages, 2 figures, 1 table Minor errors in the numerical expample, fixed

We extend the work of T. Lyons [T.J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (2) (1998) 215–310] and T. Lyons and Z. Qian [T. Lyons, Z. Qian, System Control and Rough Paths, Oxford Math. Monogr. Oxford Univ. Press, Oxford, 2002] to define integrals and solutions of differential equations along product of p and q rough paths, with 1/p +1/q >1. We use this to write an Itô formula at the level of rough paths, and to see that any rough path can always be interpreted as a product of a p-geometric rough path and a p/2-geometric rough path.

It is well known that there is a mathematical equivalence between ‘solving’ parabolic partial differential equations (PDEs) and ‘the integration’ of certain functionals on Wiener space. Monte Carlo simulation of stochastic differential equations (SDEs) is a naive approach based on this underlying principle. In finite dimensions, it is well known that cubature can be a very effective approach to integration. We discuss the appropriate extension of this idea to Wiener space. In the process we develop high–order numerical schemes valid for high–dimensional SDEs and semi–elliptic PDEs.

Fractional Sobolev spaces, also known as Besov or Slobodetzki spaces, arise in many areas of analysis, stochastic analysis in particular. We prove an embedding into certain q-variation spaces and discuss a few applications. First we show q-variation regularity of Cameron-Martin paths associated to fractional Brownian motion and other Volterra processes. This is useful, for instance, to establish large deviations for enhanced fractional Brownian motion. Second, the q-variation embedding, combined with results of rough path theory, provides a different route to a regularity result for stochastic differential equations by Kusuoka. Third, the embedding theorem works in a non-commutative setting and can be used to establish Hoelder/variation regularity of rough paths.