General order conditions for stochastic partitioned Runge-Kutta methods

In this paper stochastic partitioned Runge-Kutta (SPRK) methods are considered. A general order theory for SPRK methods based on stochastic B-series and multicolored, multishaped rooted trees is developed. The theory is applied to prove the order of some known methods, and it is shown how the number of order conditions can be reduced in some special cases, especially that the conditions for preserving quadratic invariants can be used as simplifying assumptions

Symplectic methods for Hamiltonian systems with additive noise

Stochastic systems, phase flows of which have integral invariants, are considered. Hamiltonian systems with additive noise being a wide class of such systems possess the property of preserving symplectic structure. For them, numerical methods preserving the symplectic structure are constructed. A special attention is paid to systems with separable Hamiltonians, to second order differential equations with additive noise, and to Hamiltonian systems with small additive noise

Variational Integrators and Generating Functions for Stochastic Hamiltonian Systems

In this work, the stochastic version of the variational principle is established, important for stochastic symplectic integration, and for structure-preserving algorithms of stochastic dynamical systems. Based on it, the stochastic variational integrators in formulation of stochastic Lagrangian functions are proposed, and some applications to symplectic integrations are given. Three types of generating functions in the cases of one and two noises are discussed for constructing new schemes