Sobolev weak solutions of the Hamilton--Jacobi--Bellman equations

This paper is concerned with the Sobolev weak solutions of the Hamilton-Jacobi-Bellman (HJB) equations. These equations are derived from the dynamic programming principle in the study of stochastic optimal control problems. Adopting the Doob-Meyer decomposition theorem as one of the main tools, we prove that the optimal value function is the unique Sobolev weak solution of the corresponding HJB equation. In the recursive optimal control problem, the cost function is described by the solution of a backward stochastic differential equation (BSDE). This problem has a practical background in economics and finance. We prove that the value function is the unique Sobolev weak solution of the related HJB equation by virtue of the nonlinear Doob-Meyer decomposition theorem introduced in the study of BSDEs.