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Unformatted text preview: Numerical Methods for Nonlinear Stochastic Differential Equations with Jumps * Desmond J. Higham â€ Peter E. Kloeden â€¡ AMS Subject Classification: 65C30, 65L20, 60H10 Key words: Astability, Bstability, backward Euler, compensated Poisson process, Eulerâ€“ Maruyama, exponential stability, global Lipschitz, implicit method, jumpdiffusion, mean square stability, nonlinear stability, onesided Lipschitz, Poisson process, strong convergence. Abstract We present and analyse two implicit methods for Ito stochastic differential equations (SDEs) with Poissondriven jumps. The first method, SSBE, is a splitstep extension of the backward Euler method. The second method, CSSBE, arises from the introduction of a compensated, martingale, form of the Poisson process. We show that both methods are amenable to rigorous analysis when a onesided Lipschitz condition, rather than a more restrictive global Lipschitz condition, holds for the drift. Our analysis covers strong convergence and nonlinear stability. We prove that both methods give strong convergence when the drift coefficient is onesided Lipschitz and the diffusion and jump coefficients are globally Lipschitz. On the way to proving these results, we show that a compensated form of the Eulerâ€“Maruyama method converges strongly when the SDE coefficients satisfy a local Lipschitz condition and the p th moment of the exact and numerical solution are bounded for some p > 2. Under our assumptions, both SSBE and CSSBE give welldefined, unique solutions for sufficiently small stepsizes, and SSBE has the advantage that the restriction is independent of the jump intensity. We also study the ability of the methods to reproduce exponential meansquare stability in the case where the drift has a negative onesided Lipschitz constant. This work extends the deterministic nonlinear stability theory in numerical analysis. We find that SSBE preserves stability under a stepsize constraint that is independent of the initial data. CSSBE satisfies an even stronger condition, and gives a generalization of Bstability. Finally, we specialize to a linear test problem and show that CSSBE has a natural extension of deterministic Astability. The difference in stability properties of the SSBE and CSSBE methods emphasizes that the addition of a jump term has a significant effect that cannot be deduced directly from the nonjump literature. * This manuscript appears as University of Strathclyde Mathematics Research Report 13 (2004). â€ Department of Mathematics, University of Strathclyde, Glasgow, G1 1XH, Scotland, UK ( [email protected] ). This work was supported by Engineering and Physical Sciences Research Council grant GR/T19100 and by a Research Fellowship from The Royal Society of Edinburgh/Scottish Executive Education and Lifelong Learning Department....
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This note was uploaded on 12/14/2011 for the course MAD 5932 taught by Professor Gallivan during the Fall '06 term at FSU.
 Fall '06
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