Representation of solutions to Hamilton-Jacobi-Bellman equations using constrained backward stochastic differential equations

The aim of this work is to provide, following the paper [1], a representation formula for the Hamilton-Jacobi-Bellman equation using a Forward Backward Stochastic Differential Equations (FBSDE) system. In order to do this, we introduce a class of BSDE where the generator and the final condition adapted to a bigger filtration than the one generated by the Brownian motion. This equation can be reformulated as a BSDE with constraints on the gain process. We discuss the existence and uniqueness of a minimal solution to this BSDE under reasonable assumptions. We show how this class of equations, when the generator and terminal data are given by a forward diffusion process, provides a representation formula for the solution (in a viscosity sense) to HJB fully non-linear PDE arising in stochastic control problems. In addition, as it is done in [1], we introduce an auxiliary dual control problem to which the solution to the BSDE provides the optimal value. This implies a dual representation for stochastic control problems as both are represented by the solutionto the BSDE. [1] Idris Kharroubi and Huyên Pham. ‘Feynman-Kac representation for Hamilton-Jacobi-Bellman IPDE’. In: arXiv preprint arXiv:1212.2000, to appear on Annals Of Probability (2012)