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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.

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.

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