Optimal Convergence Rate of $θ$--Maruyama Method for Stochastic Volterra Integro-Differential Equations with Riemann--Liouville Fractional Brownian Motion (Advances in Applied Mathematics and Mechanics, 2021)
This paper mainly considers the optimal convergence analysis of the $θ$--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 $θ$--Maruyama method for the SVIDEs can achieve strong first-order convergence. In particular, when the $θ$--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.