Good rough path sequences and applications to anticipating stochastic calculus

Abstract

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.

On rough differential equations

“…We now state a continuity result, which improves the results on [17,16,10] for the continuity with respect to the signal, and the results from [3,13] on the continuity with respect to the vector fields.…”

“…In addition, we show that the Itô map (z 0 , f, x) → z is locally Lipschitz in all its arguments (a slightly weaker statement on the continuity with respect to the vector field was proved in [3,9] and [13], and on the Lipschitz continuity with respect to x in [16,1]) under the more stringent assumption that f is twice differentiable with γ-Hölder continuous second-order derivative with 2 + γ > p. The proof relies on some estimates on the distance between two solutions of rough differential equations, which can be also used as a priori estimates. From this, we deduce an alternative proof of the uniqueness of the solution of a differential equation controlled by a rough paths of finite p-variation with p < 3.…”

Differential Equations Driven by Rough Paths: An Approach via Discrete Approximation

“…Next we fix a smooth non-negative function φ supported on the interval [1,2] such that φ = 1, and define D * (y) = 2 −r D (yu)φ(u)du for y ≥ 1, and A * similarly. Then 2 −rD (y) ≤ D * (y) ≤D(y) with a similar inequality for A * .…”

“…The approach to (1) in [7] mirrors the standard approach to ODE's by writing them as integral equations and using Picard iteration or a contraction mapping argument. So one writes y i (t) = y i 0 + t 0 f i j (y(s))dx j (s) and the problem then is to interpret the integral on the right.…”

Curvilinear Integrals Along Enriched Paths

“…Many authors have studied the fBM with Hurst index H > 1/4, recent publications are [3,4,5,6,8,9,10,12,13]. Let X t be the IR d -valued linear standard Brownian motion.…”