The stochastic maximum principle in optimal control of degenerate diffusions with non-smooth coefficients

For a controlled stochastic differential equation with a finite horizon cost functional, a necessary condition for optimal control of degenerate diffusions with non-smooth coefficients is derived. The main idea is to show that the SDE admits a unique linearized version interpreted as its distributional derivative with respect to the initial condition. We use a technique of Bouleau-Hirsch on absolute continuity of probability measures in order to define the adjoint process on an extension of the initial probability space.