A Forward-backward Algorithm for Stochastic Control Problems - Using the Stochastic Maximum Principle as an Alternative to Dynamic Programming

An algorithm for solving continuous-time stochastic optimal control problems is presented. The numerical scheme is based on the stochastic maximum principle (SMP) as an alternative to the widely studied dynamic programming principle (DDP). By using the SMP, (Peng, 1990) obtained a system of coupled forwardbackward stochastic differential equations (FBSDE) with an external optimality condition. We extend the numerical scheme of (Delarue and Menozzi, 2006) by a Newton-Raphson method to solve the FBSDE system and the optimality condition simultaneously. As far as the authors are aware, this is the first fully explicit numerical scheme for the solution of optimal control problems through the solution of the corresponding extended FBSDE system. We discuss possible numerical advantages to the DDP approach and consider an optimal investment-consumption problem as an example.