An Euler–Maruyama method for Caputo–Hadamard fractional stochastic differential equations on exponential meshes and its fast approximation

Abstract

This paper studies the numerical solutions of Caputo–Hadamard fractional stochastic differential equations. Firstly, we construct an Euler–Maruyama (EM) scheme for the equations, and the corresponding convergence rate is investigated. Secondly, we propose a fast EM scheme based on the sum-of-exponentials approximation to decrease the computational cost of the EM scheme. More concretely, the fast EM scheme reduces the computational cost from \(O(N^2)\) to \(O(N\log ^2 N)\) and the storage from O(N) to \(O(\log ^2 N)\) when the final time \(T\approx e\), where N is the total number of time steps. Moreover, considering the statistical errors from Monte Carlo path approximations, multilevel Monte Carlo (MLMC) techniques are utilized to reduce computational complexity. In particular, for a prescribed accuracy \(\varepsilon >0\), the EM scheme and the fast EM scheme, integrated with the MLMC technique, respectively reduce the standard EM scheme’s computational complexity from \(O(\varepsilon ^{-2-\frac{2}{\widetilde{\alpha }}})\) to \(O(\varepsilon ^{-\frac{2}{\widetilde{\alpha }}})\) and the fast EM scheme’s complexity to \(O(\varepsilon ^{-\frac{1}{\widetilde{\alpha }}}\left|\log \varepsilon \right|^3)\), where \(0<\widetilde{\alpha }=\alpha -\frac{1}{2}<\frac{1}{2}\). Finally, numerical examples are included to verify the theoretical results and demonstrate the performance of our methods.