Singular stochastic Volterra integral equations with Mittag–Leffler kernels: well-posedness and strong convergence of θ-Maruyama method

Abstract

This paper focusses on the singular stochastic Volterra integral equations with Mittag–Leffler kernels. Some qualitative properties of the solution are given under local Lipschitz condition, which include uniqueness and existence, boundedness of pth moments, Hölder continuity and continuous dependence on the initial value. The θ-Maruyama method is proposed for solving the equations. The strong convergence results of this method are obtained under global Lipschitz condition and local Lipschitz condition, respectively. Some numerical examples are given to verify the theoretical results.

Keywords:

• Singular stochastic Volterra integral equations
• Mittag–Leffler kernels
• Well-posedness
• θ-Maruyama method
• strong convergence

Mathematics Subject Classifications:

• 45D05
• 45G05
• 60H20
• 65C30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is supported by the National Natural Science Foundation of China (No. 12071403) and the Research Foundation of Education Department of Hunan Province of China (No. 21A0108).