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

Ziheng Chen; Siqing Gan; Xiaojie Wang

doi:10.3934/dcdsb.2019154

Article Contents

2019, Volume 24, Issue 8: 4513-4545. Doi: 10.3934/dcdsb.2019154

This issue Previous Article On the Alekseev-Gröbner formula in Banach spaces Next Article Nonlocal hyperbolic population models structured by size and spatial position: Well-posedness

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

School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China

^* Corresponding author: x.j.wang7@csu.edu.cn; x.j.wang7@gmail.com(Xiaojie Wang)
^* Corresponding author: x.j.wang7@csu.edu.cn; x.j.wang7@gmail.com(Xiaojie Wang)

Received: October 31, 2017

Revised: February 28, 2019

Early access: June 2019

Published: July 2019

This work was supported by NSF of China (11571373, 11671405, 91630312), NSF of Hunan Province (2016JJ3137), Innovation-Driven Project of CSU (2017CX017), Shenghua Yuying Program of CSU and Hunan Provincial Innovation Foundation For Postgraduate (CX2018B051)

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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.

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Mathematics Subject Classification: Primary: 60H10, 60H35; Secondary: 65C50.

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Figure 1. Mean-square convergence rates for (87) (left) and (88) (right)

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Table 1. CPU time of the tamed and sine methods with different stepsizes

$h$	CPU time (second)
	non-additive case		additive case
	tamed method	sine method	tamed method	sine method
$2^{-8}$	0.869440	0.744448	0.794577	0.632519
$2^{-9}$	1.203300	0.932720	1.031226	0.776562
$2^{-10}$	1.625105	1.100245	1.276387	0.906966
$2^{-11}$	2.951017	1.887072	2.355608	1.433305
$2^{-12}$	5.789335	3.588197	4.325145	2.473830

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