Search

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 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.

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

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.

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.