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We present reduction and reconstruction procedures for the solutions ofsymmetric stochastic differential equations, similar to those available forordinary differential equations. Additionally, we use the local tangent-normaldecomposition, available when the symmetry group is proper, to construct localskew-product splittings in a neighborhood of any point in the open and denseprincipal orbit type. The general methods introduced in the first part of thepaper are then adapted to the Hamiltonian case, which is studied with specialcare and illustrated with several examples. The Hamiltonian category deserves aseparate study since in that situation the presence of symmetries implies inmost cases the existence of conservation laws, mathematically described viamomentum maps, that should be taken into account in the analysis.

REDUCTION, RECONSTRUCTION, AND SKEW-PRODUCT DECOMPOSITION OF SYMMETRIC STOCHASTIC DIFFERENTIAL EQUATIONS