Generalized Hamilton-Jacobi-Bellman equations with Dirichlet boundary and stochastic exit time optimal control problem

TL;DR

This paper investigates stochastic exit time optimal control problems with nonlinear cost functions, establishing the regularity of the value function, proving the dynamic programming principle, and showing it as a viscosity solution to a generalized Hamilton-Jacobi-Bellman equation with Dirichlet boundary conditions.

Contribution

It extends Peng's backward semigroup method to analyze the regularity and viscosity solutions of generalized HJB equations with boundary conditions for stochastic exit time control problems.

Findings

Proved the value function's regularity for the control problem.

Established the dynamic programming principle in this context.

Demonstrated the value function as a viscosity solution to the generalized HJB equation.

Abstract

We consider a kind of stochastic exit time optimal control problems, in which the cost function is defined through a nonlinear backward stochastic differential equation. We study the regularity of the value function for such a control problem. Then extending Peng's backward semigroup method, we show the dynamic programming principle. Moreover, we prove that the value function is a viscosity solution to the following generalized Hamilton-Jacobi-Bellman equation with Dirichlet boundary: \[ \left\{ \begin{array} [c]{l} \inf\limits_{v\in V}\left\{\mathcal{L}(x,v)u(x)+f(x,u(x),\nabla u(x) \sigma(x,v),v)\right\}=0, \quad x\in D,\medskip\\ u(x)=g(x),\quad x\in \partial D, \end{array} \right. \] where $D$ is a bounded set in $\mathbb{R}^{d}$, $V$ is a compact metric space in $\mathbb{R}^{k}$, and for $u\in C^{2}(D)$ and $(x,v)\in D\times V$, \[\mathcal{L}(x,v)u(x):=\frac{1}{2}\sum_{i,j=1}^{d}(\sigma\sigma^{\ast})_{i,j}(x,v)\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}(x) +\sum_{i=1}^{d}b_{i}(x,v)\frac{\partial u}{\partial x_{i}}(x). \]