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Volume 8
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10.3390/math8122195
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Article

Simulating Stochastic Differential Equations with Conserved Quantities by Improved Explicit Stochastic Runge–Kutta Methods

by
Zhenyu Wang
,
Qiang Ma
and
Xiaohua Ding
*
Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, China
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(12), 2195; https://doi.org/10.3390/math8122195
Submission received: 11 November 2020 / Revised: 5 December 2020 / Accepted: 7 December 2020 / Published: 9 December 2020
(This article belongs to the Special Issue Advances in Stochastic Differential Equations and Applications to Finance)
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Abstract

Explicit numerical methods have a great advantage in computational cost, but they usually fail to preserve the conserved quantity of original stochastic differential equations (SDEs). In order to overcome this problem, two improved versions of explicit stochastic Runge–Kutta methods are given such that the improved methods can preserve conserved quantity of the original SDEs in Stratonovich sense. In addition, in order to deal with SDEs with multiple conserved quantities, a strategy is represented so that the improved methods can preserve multiple conserved quantities. The mean-square convergence and ability to preserve conserved quantity of the proposed methods are proved. Numerical experiments are implemented to support the theoretical results.
Keywords: stochastic differential equations; explicit stochastic Runge–Kutta methods; mean-square convergence; conserved quantities stochastic differential equations; explicit stochastic Runge–Kutta methods; mean-square convergence; conserved quantities

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MDPI and ACS Style

Wang, Z.; Ma, Q.; Ding, X. Simulating Stochastic Differential Equations with Conserved Quantities by Improved Explicit Stochastic Runge–Kutta Methods. Mathematics 2020, 8, 2195. https://doi.org/10.3390/math8122195

AMA Style

Wang Z, Ma Q, Ding X. Simulating Stochastic Differential Equations with Conserved Quantities by Improved Explicit Stochastic Runge–Kutta Methods. Mathematics. 2020; 8(12):2195. https://doi.org/10.3390/math8122195

Chicago/Turabian Style

Wang, Zhenyu, Qiang Ma, and Xiaohua Ding. 2020. "Simulating Stochastic Differential Equations with Conserved Quantities by Improved Explicit Stochastic Runge–Kutta Methods" Mathematics 8, no. 12: 2195. https://doi.org/10.3390/math8122195

APA Style

Wang, Z., Ma, Q., & Ding, X. (2020). Simulating Stochastic Differential Equations with Conserved Quantities by Improved Explicit Stochastic Runge–Kutta Methods. Mathematics, 8(12), 2195. https://doi.org/10.3390/math8122195

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