Summary: Let ; be a standard -dimensional Wiener process defined on a probability space (,, and let denote its natural filtration. Given a measurable d-dimensional random vector X, we look for an adapted pair of processes x(t), y(t); with values in and respectively, which solves an equation of the form:
A linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process. We also generalize our results to the following equation:
under rather restrictive assumptions on g.