The existence and uniqueness of viscosity solutions to generalized Hamilton-Jacobi-Bellman equations

In this paper, we study the existence and uniqueness of viscosity solutions to generalized Hamilton-Jacobi-Bellman (HJB) equations combined with algebra equations. This generalized HJB equation is related to a stochastic optimal control problem for which the state equation is described by a fully coupled forward-backward stochastic differential equation (FBSDE). By extending Peng's backward semigroup approach to this problem, we obtain the dynamic programming principle (DPP) and show that the value function is a viscosity solution to this generalized HJB equation. As for the proof of the uniqueness of viscosity solution, the analysis method in Barles, Buckdahn and Pardoux Baeles-BP usually does not work for this fully coupled case. With the help of the uniqueness of the solution to FBSDEs, we propose a novel probabilistic approach to study the uniqueness of the solution to this generalized HJB equation. We obtain that the value function is the minimum viscosity solution to this generalized HJB equation. Especially, when the coefficients are independent of the control variable or the solution is smooth, the value function is the unique viscosity solution.