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 $(x,v)\in D\times V$, \[\mathcal{ L}(x,v)u(x):=\frac{ 1}{ 2}\sum_{ i,j=1}^{ d}(\sigma\sigma^{ \ast})_{ i,j}(x,v)\frac{ \partial^{ 2}u}{ \partial x_{ i}\partial x_{ j}}(x) +\sum_{ i=1}^{ d}b_{ i}(x,v)\frac{ \partial u}{ \partial x_{ i}}(x). \]
Categories: math.PR math.OC