Path-wise solutions of SDE's driven by Levy processes

Abstract

In this paper we show that a path-wise solution to the following integral equation $$ Y_t = \int_0^t f(Y_t) dX_t \qquad Y_0=a \in \R^d $$ exists under the assumption that X_t is a Lévy process of finite p-variation for some $p \geq1$ and that f is an $α$-Lipschitz function for some alpha>p. There are two types of solution, determined by the solution's behaviour at jump times of the process X, one we call geometric the other forward. The geometric solution is obtained by adding fictitious time and solving an associated integral equation. The forward solution is derived from the geometric solution by correcting the solution's jump behaviour. Lévy processes, generally, have unbounded variation. So we must use a pathwise integral different from the Lebesgue-Stieltjes integral. When X has finite p-variation almost surely for p<2 we use Young's integral. This is defined whenever f and g have finite p and q-variation for 1/p+1/q>1 (and they have no common discontinuities). When p>2 we use the integral of Lyons. In order to use this integral we construct the Lévy area of the Lévy process and show that it has finite (p/2)-variation almost surely.