Convergence and stability of implicit methods for jump-diffusion systems
Desmond J. Higham 11 Department of Mathematics, University of Strathclyde, Glasgow, G1 1XH, Scotland, UK
2 Fachbereich Mathematik, Johan Wolfgang Goethe-Universität, D-60054 Frankfurt am Main, Germany
Received by the editors September, 2004 and, in revised form, November, 2004
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 analyzed.
AMS subject classifications: 65C30, 65L20, 60H10
Key words: A-stability; backward Euler; Euler-Maruyama; linear stability; Poisson process; stochastic differential equation; strong convergence; theta method; trapezoidal rule
Email: firstname.lastname@example.org (D. J. Higham), email@example.com (P. E. Kloeden)