Diffeomorphic flows driven by Levy processes

Abstract

We prove that the stochastic differential equation $$ Y_{s,t}(x) = Y_{s,s}(x) + \int_0^{t-s} f(Y_{s,s+u}(x)) dX_{s+u}, Y_{s,s}(x)=x\in\R^d. $$ driven by a Lévy process whose paths have finite p-variation almost surely for some $p\in[1,2)$ defines a flow of locally C^1-diffeomorphisms provided the vector field f is $α$-Lipschitz for some $α>p$. Using a path- wise approach we relax the smoothness condition normally required for a class of discontinuous semi-martingales.