A modified EM method and its fast implementation for multi-term Riemann-Liouville stochastic fractional differential equations

In this paper, a modified Euler-Maruyama (EM) method is constructed for a kind of multi-term Riemann-Liouville stochastic fractional differential equations and the strong convergence order min{1-{\alpha}_m, 0.5} of the proposed method is proved with Riemann-Liouville fractional derivatives' orders 0<{\alpha}_1<{\alpha}_2<...<{\alpha}_m <1. Then, based on the sum-of-exponentials approximation, a fast implementation of the modified EM method which is called a fast EM method is derived to greatly improve the computational efficiency. Finally, some numerical examples are carried out to support the theoretical results and show the powerful computational performance of the fast EM method.