The existence and uniqueness of viscosity solution to a kind of Hamilton-Jacobi-Bellman equations

TL;DR

This paper proves the existence and uniqueness of viscosity solutions for a class of Hamilton-Jacobi-Bellman equations linked to stochastic control problems involving coupled forward-backward stochastic differential equations, using a novel probabilistic approach.

Contribution

It extends the theory of viscosity solutions to fully coupled HJB equations with a new probabilistic method for proving uniqueness.

Findings

The value function is a viscosity solution to the HJB equation.

The value function is the minimal viscosity solution.

Uniqueness holds when coefficients are control-independent or solutions are smooth.

Abstract

In this paper, we study the existence and uniqueness of viscosity solutions to a kind of Hamilton-Jacobi-Bellman (HJB) equations combined with algebra equations. This 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. By extending Peng's backward semigroup approach to this problem, we obtain the dynamic programming principle and show that the value function is a viscosity solution to this HJB equation. As for the proof of the uniqueness of viscosity solution, the analysis method in Barles, Buckdahn and Pardoux <cite>Baeles-BP</cite> 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 HJB equation. We obtain that the value function is the minimum viscosity solution to this 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.