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2019, Volume 24, Issue 10: 5553-5567. Doi: 10.3934/dcdsb.2019071

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Existence and approximation of strong solutions of SDEs with fractional diffusion coefficients

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

¹ Corresponding author
¹ Corresponding author

Received: July 2018

Revised: October 2018

Early access: April 2019

Published: August 2019

The research was supported in part by the National Natural Science Foundations of China (Grant Nos. 61473125 and 11761130072) and the Royal Society-Newton Advanced Fellowship (REF NA160317).

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Abstract

In stochastic financial and biological models, the diffusion coefficients often involve the terms $ \sqrt{|x|} $ and $ \sqrt{|x(1-x)|} $, or more general $ |x|^{r} $ and $ |x(1-x)|^r $ for $ r $ $ \in $ $ (0, 1) $. These coefficients do not satisfy the local Lipschitz condition, which implies that the existence and uniqueness of the solution cannot be obtained by the standard conditions. This paper establishes the existence and uniqueness of the strong solution and the strong convergence of the Euler-Maruyama approximations under certain conditions for systems of stochastic differential equations for which one component has such a diffusion coefficient with $ r $ $ \in $ $ [1/2, 1) $.

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Mathematics Subject Classification: Primary: 60H10; Secondary: 65C05.

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