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A New Explicit Magnus Expansion for Nonlinear Stochastic Differential Equations

1
Department of Mathematics,Tianjin Chengjian University, Tianjin 300384, China
2
Department of Mathematics, Tongji University, Shanghai 200092, China
3
Business School, University of Shanghai for Science and Technology, Shanghai 200092, China
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(2), 183; https://doi.org/10.3390/math8020183
Received: 2 January 2020 / Revised: 21 January 2020 / Accepted: 22 January 2020 / Published: 3 February 2020
(This article belongs to the Section Computational Mathematics)
In this paper, based on the iterative technique, a new explicit Magnus expansion is proposed for the nonlinear stochastic equation d y = A ( t , y ) y d t + B ( t , y ) y d W . One of the most important features of the explicit Magnus method is that it can preserve the positivity of the solution for the above stochastic differential equation. We study the explicit Magnus method in which the drift term only satisfies the one-sided Lipschitz condition, and discuss the numerical truncated algorithms. Numerical simulation results are also given to support the theoretical predictions.
Keywords: explicit Magnus expansion; asymptotic stability; Stratonovich integral; Itô integral; nonlinear stochastic equations explicit Magnus expansion; asymptotic stability; Stratonovich integral; Itô integral; nonlinear stochastic equations
MDPI and ACS Style

Wang, X.; Guan, X.; Yin, P. A New Explicit Magnus Expansion for Nonlinear Stochastic Differential Equations. Mathematics 2020, 8, 183.

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