A class of new Magnus-type methods for semi-linear non-commutative Itô stochastic differential equations

Abstract

In this paper, a class of new Magnus-type methods is proposed for non-commutative Ito stochastic differential equations (SDEs) with semi-linear drift term and semi-linear diffusion terms, based on Magnus expansion for non-commutative linear SDEs. We construct a Magnus-type Euler method, a Magnus-type Milstein method and a Magnus-type Derivative-free method, and give the mean-square convergence analysis of these methods. Numerical tests are carried out to present the efficiency of the proposed methods compared with the corresponding underlying methods and the specific performance of the simulation Ito integral algorithms is investigated.