Symplectic conditions and stochastic generating functions of stochastic Runge–Kutta methods for stochastic Hamiltonian systems with multiplicative noise

This paper develops numerical approximate algorithms for the solutions of stochastic Hamiltonian systems with multiplicative noise. Some conditions are captured to guarantee that a given stochastic Runge–Kutta method is symplectic. This paper also shows that stochastic symplectic Runge–Kutta methods can be written in terms of stochastic generating functions. Based on the order and symplectic conditions, some low-stage stochastic symplectic Runge–Kutta methods with strong global order 1.0 are constructed. Some numerical results are presented to demonstrate the efficiency of stochastic symplectic Runge–Kutta methods.

► Symplectic conditions of stochastic RK methods are given.
► Stochastic generating functions of stochastic symplectic RK methods are obtained.
► Some low-stage stochastic symplectic RK methods are constructed.