Rate of convergence of Euler approximation of time-dependent mixed SDEs driven by Brownian motions and fractional Brownian motions

Weiguo Liu; Yan Jiang; Zhi Li; Weiguo Liu; Yan Jiang; Zhi Li

doi:10.3934/math.2020144

AIMS Mathematics

2020, Volume 5, Issue 3: 2163-2195. doi: 10.3934/math.2020144

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Research article

Rate of convergence of Euler approximation of time-dependent mixed SDEs driven by Brownian motions and fractional Brownian motions

1.
School of Statistics and Mathematics, Guangdong University of Finance & Economics, 21 Luntou Road, Guangzhou, Guangdong, MO 510320, P. R. China
2.
School of Information and Mathematics, Yangtze University, Jingzhou, Hubei, MO 434023, P. R. China

Received: 22 October 2019 Accepted: 02 February 2020 Published: 27 February 2020
MSC : 41A25, 60G22, 60H10

A kind of time-dependent mixed stochastic differential equations driven by Brownian motions and fractional Brownian motions with Hurst parameter $H>\frac{1}{2}$ is considered. We prove that the rate of convergence of Euler approximation of the solutions can be estimated by $O(\delta^{\frac{1}{2}\wedge(2H-1)})$ in probability, where $\delta$ is the diameter of the partition used for discretization.
- Brownian motion,
- fractional Brownian motion,
- Euler approximation,
- rate of convergence
Citation: Weiguo Liu, Yan Jiang, Zhi Li. Rate of convergence of Euler approximation of time-dependent mixed SDEs driven by Brownian motions and fractional Brownian motions[J]. AIMS Mathematics, 2020, 5(3): 2163-2195. doi: 10.3934/math.2020144

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Abstract

A kind of time-dependent mixed stochastic differential equations driven by Brownian motions and fractional Brownian motions with Hurst parameter $H>\frac{1}{2}$ is considered. We prove that the rate of convergence of Euler approximation of the solutions can be estimated by $O(\delta^{\frac{1}{2}\wedge(2H-1)})$ in probability, where $\delta$ is the diameter of the partition used for discretization.

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