Fast Euler–Maruyama method for weakly singular stochastic Volterra integral equations with variable exponent

In this paper, we consider the weakly singular stochastic Volterra integral equations with variable exponent. Firstly, the existence and uniqueness of the equations are studied by the Banach contraction mapping principle. Secondly, we develop an Euler–Maruyama (EM) method and obtain its strong convergence rate. Moreover, we propose a fast EM method via the exponential-sum-approximation technique to reduce the EM method’s computational cost. More specifically, if one disregards the Monte Carlo sampling error, then the fast EM method reduces the computational cost from O(N2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(N^2)$$\end{document} to O(Nlog2N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(N\log ^{2} N)$$\end{document} and the storage from O(N) to O(log2N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\log ^{2} N)$$\end{document}, where N is the total number of time steps. Moreover, if the sampling error is taken into account, we employ the multilevel Monte Carlo method based on the fast EM method to reduce computational costs further. Significantly, the computational costs of the EM method and the fast EM method to achieve an accuracy of O(ε) (ε < 1) are reduced from O(ε-2-2α~)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\varepsilon ^{-2-\frac {2}{\widetilde {\alpha }}})$$\end{document} and O(ε-2-1α~log2(ε))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\varepsilon ^{-2-\frac {1}{\widetilde {\alpha }}}\log ^{2}(\varepsilon ))$$\end{document}, respectively, to O(ε-1α~(log(ε-1))3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O\Big (\varepsilon ^{-\frac {1}{\widetilde {\alpha }}} (\log (\varepsilon ^{-1}))^{3}\Big )$$\end{document}, where α~=min{1-α*,12-β*}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde {\alpha }=\min \limits \{1-\alpha ^{\ast }, \frac 12-\beta ^{\ast }\}$$\end{document} is related to the exponents of the singular kernel in the equations. Finally, numerical examples are provided to illustrate our theoretical results and demonstrate the superiority of the fast EM method.