  In this paper we present splitting methods which are based on iterative
schemes and applied to stochastic nonlinear Schroedinger equation. We will
design stochastic integrators which almost conserve the symplectic structure.
The idea is based on rewriting an iterative splitting approach as a successive
approximation method based on a contraction mapping principle and that we have
an almost symplectic scheme. We apply a stochastic differential equation, that
we can decouple into a deterministic and stochatic part, while each part can be
solved analytically. Such decompositions allow accelerating the methods and
preserving, under suitable conditions, the symplecticity of the schemes. A
numerical analysis and application to the stochastic Schroedinger equation are
presented.
