Series Solution of Stochastic HJB Equations

We consider infinite horizon, stochastic, smooth optimal control problems in continuous time where the coefficients of the white Gaussian noise terms in the dynamics vanish at the origin. We show how the Taylor polynomials of the optimal cost and the optimal feedback can be computed degree by degree. This is a generaliztion of the work of Al'brekht who showed how the Taylor polynomials of the optimal cost and the optimal feedback for deterministic problems can be computed degree by degree.