  We present reduction and reconstruction procedures for the solutions of
symmetric stochastic differential equations, similar to those available for
ordinary differential equations. Additionally, we use the local tangent-normal
decomposition, available when the symmetry group is proper, to construct local
skew-product splittings in a neighborhood of any point in the open and dense
principal orbit type. The general methods introduced in the first part of the
paper are then adapted to the Hamiltonian case, which is studied with special
care and illustrated with several examples. The Hamiltonian category deserves a
separate study since in that situation the presence of symmetries implies in
most cases the existence of conservation laws, mathematically described via
momentum maps, that should be taken into account in the analysis.
