Stochastic symplectic partitioned Runge–Kutta methods for stochastic Hamiltonian systems with multiplicative noise

• A new class of SPRK methods and their order conditions are given.
• Symplectic conditions of the SPRK methods are obtained.
• Stochastic generating functions of the SSPRK methods are obtained.
• It is proved that the SSPRK methods can preserve the quadratic invariants.
• Some low-stage SSPRK methods are constructed.

Some new stochastic partitioned Runge–Kutta (SPRK) methods are proposed for the strong approximation of partitioned stochastic differential equations (SDEs). The order conditions up to strong global order 1.0 are calculated. The SPRK methods are applied to solve stochastic Hamiltonian systems with multiplicative noise. Some conditions are captured to guarantee that a given SPRK method is symplectic. It is shown that stochastic symplectic partitioned Runge–Kutta (SSPRK) methods can be written in terms of stochastic generating functions. In addition, this paper also proves that the SSPRK methods can conserve the quadratic invariants of original stochastic systems. Based on the order and symplectic conditions, some low-stage SSPRK methods with strong global order 1.0 are constructed. Finally, some numerical results are presented to demonstrate the efficiency of the SSPRK methods.