Convergence and Stability of Implicit Methods for Jump-Diffusion Systems

Abstract

A class of implicit methods is introduced for Ito stochastic difference equations with Poisson-driven jumps. A convergence proof shows that these implicit methods share the same finite time strong convergence rate as the explicit Euler-Maruyama scheme. A mean-square linear stability analysis shows that implicitness offers benefits, and a natural analogue of mean-square A-stability is studied. Weak variants are also considered and their stability is analyzed.