A new class of structure-preserving stochastic exponential Runge-Kutta integrators for stochastic differential equations

In this article, a new class of stochastic exponential Runge-Kutta (SERK) methods is developed for solving stochastic differential equations. The proposed SERK methods can preserve conformal quadratic invariants and conformal symplectic structure automatically under certain coefficient conditions. Stochastic B-series theory is generalized, which allows the study of the mean-square convergence order conditions of the SERK methods. Some low stage stochastic exponential integrators with 1 order mean-square convergence and structure-preserving properties are given. For damped Hamiltonian systems with additive noise terms, a class of stochastic exponential integrators with 1.5 order mean-square convergence and conformal symplectic structure preservation is constructed. Numerical tests show the efficacy of the stochastic exponential integrators.