Stochastic control of systems with control multiplicative noise using second order FBSDEs

The Hamilton Jacobi Bellman (HJB) PDE for the stochastic optimal control (SOC) problem for diffusion SDE dynamics which have affine controls and state and control multiplicative noise is a second order fully nonlinear PDE. The previously known linearly solvable optimal control framework as well as the first order forward backward SDEs (FBSDEs) frameworks therefore have a characteristic inadequacy to support sampling algorithms for this SOC problem. We present the framework of second order FBSDEs for solving this SOC problem in this paper. We derive the nonlinear Feynman Kac representation of the second order fully nonlinear HJB PDE corresponding to diffusions with state and control multiplicative noise. The Feynman Kac representation enables a sampling based scheme for solving this SOC problem. This scheme is then leveraged to develop a least squares Monte Carlo regression based algorithm for implementations. The algorithm is validated by examples of simulated control of an underactuated system and comparison against an analytical characterization.