  This paper proves a version for stochastic differential equations of the
Lie-Scheffers Theorem. This result characterizes the existence of nonlinear
superposition rules for the general solution of those equations in terms of the
involution properties of the distribution generated by the vector fields that
define it. When stated in the particular case of standard deterministic
systems, our main theorem improves various aspects of the classical
Lie-Scheffers result. We show that the stochastic analog of the classical
Lie-Scheffers systems can be reduced to the study of Lie group valued
stochastic Lie-Scheffers systems; those systems, as well as those taking values
in homogeneous spaces are studied in detail. The developments of the paper are
illustrated with several examples.

Probability;Differential Geometry