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Int. J. Numer. Anal. Mod., 3 (2006), pp. 125-140. |
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Convergence and stability of implicit methods for jump-diffusion systems Desmond J. Higham 1 1 Department of Mathematics, University of Strathclyde, Glasgow, G1 1XH, Scotland, UK2 Fachbereich Mathematik, Johan Wolfgang Goethe-Universität, D-60054 Frankfurt am Main, Germany Received by the editors September, 2004 and, in revised form, November, 2004 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 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: djh@maths.strath.ac.uk (D. J. Higham), kloeden@math.uni-frankfurt.de (P. E. Kloeden) |