Well-posedness, Regularity of Solutions and the \(\theta \)-Euler-Maruyama Scheme for Stochastic Volterra Integral Equations with General Singular Kernels and Jumps

Abstract

In this paper, we consider a class of stochastic Volterra integral equations with general singular kernels, driven by a Brownian motion and a pure jump Lévy process. We first show that these equations have a unique strong solution under certain regular conditions on their coefficients. Furthermore, the solutions of this equation depend continuously on the initial value and on the kernels k, \(k_B\), and \(k_Z\). We will then show the regularity of solutions for these equations. Finally, we propose a \(\theta \)-Euler-Maruyama approximation scheme for these equations and demonstrate its convergence at a certain rate in the \(L^2\)-norm. Some numerical simulations is also presented to support for the theoretical results.