Mean-square symplectic methods for Hamiltonian systems with multiplicative noise

Stochastic systems with multiplicative noise, phase flows of which have integral invariants, are considered. For such systems, numerical methods preserving the integral invariants are constructed using full implicit schemes of a new type for stochastic differential equations. In these full implicit schemes increments of Wiener processes are substituted by some truncated random variables. They are important for both theory and practice of numerical integration of stochastic differential equations. A special attention is paid to systems with separable Hamiltonians and to Hamiltonian systems with small noise. Liouvillian methods for stochastic systems preserving phase volume are also proposed. Some results of numerical experiments are presented