Stochastic differential equations with path-independent solutions

Imme van den Berg

Published 2012-09-04Version 1

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.