Numerical methods for nonlinear stochastic differential equations with jumps.

*(English)*Zbl 1186.65010Summary: 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.

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 |

##### Keywords:

A-stability; B-stability; backward Euler; compensated Poisson process; Euler - Maruyama; exponential stability; global Lipschitz; implicit method; jump-diffusion; mean-square stability; nonlinear stability; one-sided Lipschitz; Poisson process; strong convergence; stepsize control; numerical examples
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\textit{D. J. Higham} and \textit{Peter. E. Kloeden}, Numer. Math. 101, No. 1, 101--119 (2005; Zbl 1186.65010)

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