Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler--Maruyama approximation

Abstract

This paper considers the initial value problem of general nonlinear stochastic fractional integro-differential equations with weakly singular kernels. Our effort is devoted to establishing some fine estimates to include all the cases of Abel-type singular kernels. Firstly, the existence, uniqueness and continuous dependence on the initial value of the true solution under local Lipschitz condition and linear growth condition are derived in detail. Secondly, the Euler--Maruyama method is developed for solving numerically the equation, and then its strong convergence is proven under the same conditions as the well-posedness. Moreover, we obtain the accurate convergence rate of this method under global Lipschitz condition and linear growth condition. In particular, the Euler--Maruyama method can reach strong first-order superconvergence when $\alpha =1$. Finally, several numerical tests are reported for verification of the theoretical findings.

References in corpus (5)

• Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients
• Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients
• Fractional Integro-Differential Equations for Electromagnetic Waves in Dielectric Media
• Asymptotic separation between solutions of Caputo fractional stochastic differential equations
• Fractional stochastic differential equations satisfying fluctuation-dissipation theorem

Cited by in corpus (2)

• Mittag--Leffler Euler integrator and large deviations for stochastic space-time fractional diffusion equations
• Euler-Maruyama approximation for stochastic fractional neutral integro-differential equations with weakly singular kernel