Resumen de Path-wise solutions of stochastic differential equations driven by Lévy processes

D. R. E. Williams

• In this paper we show that a path-wise solution to the following integral equation Yt = ?0t f(Yt) dXt, Y0 = a Î Rd, exists under the assumption that Xt is a Lévy process of finite p-variation for some p = 1 and that f is an a-Lipschitz function for some a > p. We examine 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 a 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. 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.