Mean-square approximations of Lévy noise driven SDEs with super-linearly growing diffusion and jump coefficients

Abstract

This paper first establishes a fundamental mean-square convergence theorem for general one-step numerical approximations of Lévy noise driven stochastic differential equations with non-globally Lipschitz coefficients. Then two novel explicit schemes are designed and their convergence rates are exactly identified via the fundamental theorem. Different from existing works, we do not impose a globally Lipschitz condition on the jump coefficient but formulate appropriate assumptions to allow for its super-linear growth. However, we require that the Lévy measure is finite. New arguments are developed to handle essential difficulties in the convergence analysis, caused by the super-linear growth of the jump coefficient and the fact that higher moment bounds of the Poisson increments $ \int_t^{t+h} \int_Z \,\bar{N}(\mbox{d}s,\mbox{d}z), t \geq 0, h >0$ contribute to magnitude not more than $O(h)$. Numerical results are finally reported to confirm the theoretical findings.

arXiv comment34pages, 2 figures

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Cited by in corpus (4)

• Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities
• Well-posedness and tamed Euler schemes for McKean-Vlasov equations driven by Lévy noise
• A stochastic Gronwall inequality and applications to moments, strong completeness, strong local Lipschitz continuity, and perturbations
• On the backward Euler method for a generalized Ait-Sahalia-type rate model with Poisson jumps