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Well-posedness, Regularity of Solutions and the $\theta$-Euler-Maruyama Scheme for Stochastic Volterra Integral Equations with General Singular Kernels and Jumps
Phan Thi Huong
,
Hoang-Long Ngo
,
Peter Kloeden
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.