Compensated split-step balanced methods for nonlinear stiff SDEs with jump-diffusion and piecewise continuous arguments
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https://doi.org/10.1007/s11425-019-1781-6
Compensated split-step balanced methods for nonlinear stiff SDEs with jump-diffusion and piecewise continuous arguments Ying Xie1,2 & Chengjian Zhang1,3,∗ 1School
of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China; of Mathematics and Economics, Hubei University of Education, Wuhan 430205, China; 3Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China
2School
Email: [email protected], [email protected] Received July 25, 2019; accepted September 17, 2020
Abstract
This paper deals with numerical solutions of nonlinear stiff stochastic differential equations with
jump-diffusion and piecewise continuous arguments. By combining compensated split-step methods and balanced methods, a class of compensated split-step balanced (CSSB) methods are suggested for solving the equations. Based on the one-sided Lipschitz condition and local Lipschitz condition, a strong convergence criterion of CSSB methods is derived. It is proved under some suitable conditions that the numerical solutions produced by CSSB methods can preserve the mean-square exponential stability of the corresponding analytical solutions. Several numerical examples are presented to illustrate the obtained theoretical results and the effectiveness of CSSB methods. Moreover, in order to show the computational advantage of CSSB methods, we also give a numerical comparison with the adapted split-step backward Euler methods with or without compensation and tamed explicit methods. Keywords
stiff stochastic differential equation, jump diffusion, piecewise continuous argument, compensated
split-step balanced method, strong convergence, mean-square exponential stability MSC(2010)
65C20, 60H35, 65L20
Citation: Xie Y, Zhang C J. Compensated split-step balanced methods for nonlinear stiff SDEs with jump-diffusion and piecewise continuous arguments. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-0191781-6
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Introduction
Among stochastic differential equations (SDEs), there are a class of equations with jump-diffusion (JDSDEs). In the recent years, this class of equations have attracted an increasing interest due to their effectiveness in modeling some uncertainty problems in control science, biology, economics and other scientific and engineering fields (see [3,6,9,10,13,18,21,35]). Nevertheless, it is difficult to obtain the explicit solutions of JDSDEs. Hence, developing various numerical methods to solve JDSDEs becomes an important topic (see [9, 10, 12–15, 17, 27, 28, 32, 39]). In the existing references, numerical methods for JDSDEs * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
math.scichina.com
link.springer.com
Xie Y et al.
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Sci China Math
with delay have been concerned by some authors. For example, Higham and Kloeden [9] constructed the compensated split-step backward Euler (CSSBE) methods and studied t
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