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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Compensated projected Euler-Maruyama method for stochastic differential equations with superlinear jumpsR Min Li a, Chengming Huang a,b,∗, Ziheng Chen c a
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China c Institute of Computational Mathematics, Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China b
a r t i c l e
i n f o
Article history: Received 29 April 2020 Revised 5 October 2020 Accepted 18 October 2020
Keywords: Stochastic differential equations with jumps Compensated projected Euler-Maruyama method Mean square convergence C-stability B-consistency
a b s t r a c t In this paper, we present and analyze a compensated projected Euler-Maruyama method for stochastic differential equations with jumps. A mean square convergence result is derived under a coupled condition. This condition and some reasonable assumptions admit that the jump and diffusion coefficients can be superlinear. Moreover, since the Poisson increment has different moment properties from the Brownian increment, some new techniques are developed for convergence analysis. Finally, some numerical experiments are carried out to confirm the theoretical results. © 2020 Elsevier Inc. All rights reserved.
1. Introduction Consider stochastic differential equations (SDEs) with Poisson-driven jumps of the form
dY (t ) = f (Y (t − ))dt + g(Y (t − ))dW (t ) + h(Y (t − ))dN (t ),
t ∈ ( 0, T ],
(1.1)
with Y (0 ) = Y0 , and f, h : Rd → Rd , g : Rd → Rd×m . Here, Y (t − ) is defined by lims→t − Y (s ), W (t ) is an m-dimensional Brownian motion and N (t ) is a scalar Poisson process with intensity λ > 0. These equations have many applications in physics [1], engineering [2], and particularly in financial mathematics [3,4]. Since SDEs with jumps are hard to solve explicitly, numerical simulations are required in practice. In the past several decades, a great number of numerical methods have been developed (see [5–13] and references therein). Most of mentioned works require that the diffusion coefficients of the SDEs with jumps satisfy a global Lipschitz condition or a linear growth condition. However, many practical mathematical models do not meet such strong constraints and perhaps behave superlinearly (see [14,15]). The numerical solution of jump-diffusion SDEs with superlinear diffusion coefficients attracted attention only recently. Huang [16] introduced the compensated split-step theta (CSST) method and discussed the mean square stability of two classes of theta methods. Dareiotis et al. [17] studied tamed Euler method for a class of nonlinear SDEs driven by Lévy R ∗
This work was supported by National Natural Science Foundation of China (Nos. 11771163 and 12011530058).
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