Volume 14, Issue 1
Optimal Convergence Rate of $\theta$--Maruyama Method for Stochastic Volterra Integro-Differential Equations with Riemann--Liouville Fractional Brownian Motion

Adv. Appl. Math. Mech., 14 (2022), pp. 202-217.

Published online: 2021-11

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• Abstract

This paper mainly considers the optimal convergence analysis of the $\theta$--Maruyama method for stochastic Volterra integro-differential equations (SVIDEs) driven by Riemann--Liouville fractional Brownian motion under the global Lipschitz and linear growth conditions. Firstly, based on the contraction mapping principle, we prove the well-posedness of the analytical solutions of the SVIDEs. Secondly, we show that the $\theta$--Maruyama method for the SVIDEs can achieve strong first-order convergence. In particular, when the $\theta$--Maruyama method degenerates to the explicit Euler--Maruyama method, our result improves the conclusion that the convergence rate is $H+\frac{1}{2},$ $H\in(0,\frac{1}{2})$ by Yang et al., J. Comput. Appl. Math., 383 (2021), 113156. Finally, the numerical experiment verifies our theoretical results.

• Keywords

Stochastic Volterra integro-differential equations, Riemann--Liouville fractional Brownian motion, well-posedness, strong convergence.

65C30, 65C20, 65L20

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@Article{AAMM-14-202, author = {Wang , Mengjie and Dai , Xinjie and Xiao , Aiguo}, title = {Optimal Convergence Rate of $\theta$--Maruyama Method for Stochastic Volterra Integro-Differential Equations with Riemann--Liouville Fractional Brownian Motion}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2021}, volume = {14}, number = {1}, pages = {202--217}, abstract = {

This paper mainly considers the optimal convergence analysis of the $\theta$--Maruyama method for stochastic Volterra integro-differential equations (SVIDEs) driven by Riemann--Liouville fractional Brownian motion under the global Lipschitz and linear growth conditions. Firstly, based on the contraction mapping principle, we prove the well-posedness of the analytical solutions of the SVIDEs. Secondly, we show that the $\theta$--Maruyama method for the SVIDEs can achieve strong first-order convergence. In particular, when the $\theta$--Maruyama method degenerates to the explicit Euler--Maruyama method, our result improves the conclusion that the convergence rate is $H+\frac{1}{2},$ $H\in(0,\frac{1}{2})$ by Yang et al., J. Comput. Appl. Math., 383 (2021), 113156. Finally, the numerical experiment verifies our theoretical results.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2020-0384}, url = {http://global-sci.org/intro/article_detail/aamm/19982.html} }
TY - JOUR T1 - Optimal Convergence Rate of $\theta$--Maruyama Method for Stochastic Volterra Integro-Differential Equations with Riemann--Liouville Fractional Brownian Motion AU - Wang , Mengjie AU - Dai , Xinjie AU - Xiao , Aiguo JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 202 EP - 217 PY - 2021 DA - 2021/11 SN - 14 DO - http://doi.org/10.4208/aamm.OA-2020-0384 UR - https://global-sci.org/intro/article_detail/aamm/19982.html KW - Stochastic Volterra integro-differential equations, Riemann--Liouville fractional Brownian motion, well-posedness, strong convergence. AB -

This paper mainly considers the optimal convergence analysis of the $\theta$--Maruyama method for stochastic Volterra integro-differential equations (SVIDEs) driven by Riemann--Liouville fractional Brownian motion under the global Lipschitz and linear growth conditions. Firstly, based on the contraction mapping principle, we prove the well-posedness of the analytical solutions of the SVIDEs. Secondly, we show that the $\theta$--Maruyama method for the SVIDEs can achieve strong first-order convergence. In particular, when the $\theta$--Maruyama method degenerates to the explicit Euler--Maruyama method, our result improves the conclusion that the convergence rate is $H+\frac{1}{2},$ $H\in(0,\frac{1}{2})$ by Yang et al., J. Comput. Appl. Math., 383 (2021), 113156. Finally, the numerical experiment verifies our theoretical results.

Mengjie Wang, Xinjie Dai & Aiguo Xiao. (1970). Optimal Convergence Rate of $\theta$--Maruyama Method for Stochastic Volterra Integro-Differential Equations with Riemann--Liouville Fractional Brownian Motion. Advances in Applied Mathematics and Mechanics. 14 (1). 202-217. doi:10.4208/aamm.OA-2020-0384
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