We present a condition for a stochastic differential equation dX_{ t}={ \mu}(t,X_{ t})dt+{ \sigma}(t,X_{ t})dB_{ t} to have a unique functional solution of the form Z(t,B_{ t}). The condition expresses a relation between { \mu} and { \sigma}. A generalization concerns solutions of the form Z(t,Y_{ t}), where Y_{ t} is an Ito-process satisfying a stochastic differential equation with coefficients only depending on time, to be determined from { \mu} and { \sigma}. The solutions in question are obtained by solving a system of two partial differential equations, which may be reduced to two ordinary differential equations.
Categories: math.PR