Lévy-driven stochastic Volterra integral equations with doubly singular kernels: existence, uniqueness, and a fast EM method

This paper considers Levy noise driven nonlinear stochastic Volterra integral equations with doubly weakly singular kernels, whose singular points include both s = 0 and s = t. The existence and uniqueness theorem of the true solution as well as the strong convergence rate of the Euler–Maruyama (EM) method are developed via establishing some fine estimates. Compared with the corresponding results by Wang (Statist Probab Lett 78: 1062–1071, 2008) and Zhang (J Differential Equations 244: 2226–2250, 2008), our results generalize the Gaussian noise case to the Levy noise case, relax the integrable limitations of singular kernels and establish an accurate convergence order. Moreover, based on the efficient sum-of-exponentials approximation, a fast EM method is presented to improve the low computational efficiency of the EM method. Specifically, when T ≫ 1, the computational complexity O(N2) and the storage O(N) of the EM method are reduced to $O(N\log N)$ and $O(\log N)$ respectively; while T ≈ 1, they are reduced to $O(N\log ^{2} N)$ and $O(\log ^{2} N)$ respectively, where T and N denote the terminal time and the total number of time steps respectively. Finally, the numerical example supports the theoretical results and explains the priority of the fast EM method.