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Numerical methods for nonlinear stochastic differential equations with jumps. (English) Zbl 1186.65010
Summary: We present and analyse two implicit methods for Ito stochastic differential equations (SDEs) with Poisson-driven jumps. The first method, SSBE, is a split-step 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 one-sided 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 one-sided 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 Euler-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 well-defined, 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 mean-square stability in the case where the drift has a negative one-sided 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 B-stability.
Finally, we specialize to a linear test problem and show that CSSBE has a natural extension of deterministic A-stability. 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 non-jump literature.

MSC:
65C30 Numerical solutions to stochastic differential and integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
Software:
stochastic; RODAS
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[1] Baker, C.T.H., Buckwar, E.: Exponential stability in p-th mean of solutions, and of convergent Euler-type solutions, to stochastic delay differential equations. J. Computat. Appl. Math. (to appear) · Zbl 1081.65011
[2] Burrage, K., Burrage, P.M., Tian, T.: Numerical methods for strong solutions of stochastic differential equations: an overview. Proceedings: Mathematical, Physical and Engineering, Royal Society of London 460, 373–402 (2004) · Zbl 1048.65004
[3] Cont, R., Tankov, P.: Financial Modelling With Jump Processes. Chapman & Hall/CRC, Florida (2004) · Zbl 1052.91043
[4] Cyganowski, S., Grüne, L., Kloeden, P.E.: MAPLE for jump-diffusion stochastic differential equations in finance. In: Programming Languages and Systems in Computational Economics and Finance, S.S. Nielsen (ed.), Kluwer, Boston (2002), pp. 441–460
[5] Dekker, K., Verwer, J.G.: Stability of Runge–Kutta Methods for Stiff Nonlinear Equations. North Holland, Amsterdam (1984) · Zbl 0571.65057
[6] Gardoń, A.: The order of approximation for solutions of Itô-type stochastic differential equations with jumps. Stochastic Anal. Appl. 22, 679–699 (2004) · Zbl 1056.60065
[7] Gikhman, I.I., Skorokhod, A.V.: Stochastic Differential Equations. Springer-Verlag, Berlin (1972) · Zbl 0169.48702
[8] Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin, second ed. (1996) · Zbl 0859.65067
[9] Higham, D.J., Kloeden, P.E.: Convergence and stability of implicit methods for jump-diffusion systems. International Journal of Numerical Analysis & Modeling (to appear) · Zbl 1109.65007
[10] Higham, D.J., Mao, X., Stuart, A.M.: Strong convergence of Euler-like methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. 40, 1041–1063 (2002) · Zbl 1026.65003
[11] Higham, D.J., Mao, X., Stuart, A.M.: Exponential mean square stability of numerical solutions to stochastic differential equations. London Mathematical Society J. Comput. Math. 6, 297–313 (2003) · Zbl 1055.65009
[12] Hu, Y.: Semi-implicit Euler-Maruyama scheme for stiff stochastic equations. In: Stochastic Analysis and Related Topics, V; The Silivri Workshop, Progr. Probab., 38, H. Koerezlioglu, (ed.), Birkhauser, Boston (1996), pp. 183–202 · Zbl 0848.60057
[13] Maghsoodi, Y.: Mean square efficient numerical solution of jump-diffusion stochastic differential equations. Indian J. Statistics 58, 25–47 (1996) · Zbl 0881.60057
[14] Maghsoodi, Y.: Exact solutions and doubly efficient approximations and simulation of jump-diffusion Ito equations. Stochastic Anal. Appl. 16, 1049–1072 (1998) · Zbl 0920.60041
[15] Mao, X.: Stability of Stochastic Differential Equations with respect to Semimartingales. Longman Scientific and Technical, Pitman Research Notes in Mathematics Series 251 (1991) · Zbl 0724.60059
[16] Mattingly, J., Stuart, A.M., Higham, D.J.: Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise. Stochastic Processes and their Appl. 101, 185–232 (2002) · Zbl 1075.60072
[17] Milstein, G.N., Tretyakov, M.V.: Numerical integration of stochastic differential equations with nonglobally Lipschitz coefficients. SIAM J. Numer. Anal. (to appear) · Zbl 1102.60059
[18] Schurz, H.: Stability, Stationarity, and Boundedness of some Implicit Numerical Methods for Stochastic Differential Equations and Applications. Logos Verlag (1997) · Zbl 0905.60002
[19] Sobczyk, K.: Stochastic Differential Equations with Applications to Physics and Engineering. Kluwer Academic, Dordrecht (1991) · Zbl 0762.60050
[20] Stuart, A.M., Humphries, A.R.: Dynamical Systems and Numerical Analysis. Cambridge University Press, Cambridge (1996) · Zbl 0869.65043
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