Error Analysis for Approximation of Stochastic Differential Equations Driven by Poisson Random Measures

Abstract

Let Xt be the solution of a stochastic differential equation (SDE) with starting point x0 driven by a Poisson random measure. Additive functionals are of interest in various applications. Nevertheless they are often unknown and can only be found by simulation on computers. We investigate the quality of the Euler approximation. Our main emphasis is on SDEs driven by an $\alpha$-stable process, $0<\alpha < 2$, where we study the approximation of the Monte Carlo error ${\Bbb E} [f(X_T)]$, f belonging to ${L}^\infty$. Moreover, we treat the case where the time equals $T\wedge \tau$, where $\tau$ is the first exit time of some interval.