Generalized Hamilton--Jacobi--Bellman Equations with Dirichlet Boundary Condition and Stochastic Exit Time Optimal Control Problem

We consider a kind of stochastic exit time optimal control problem in which the cost functional 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 condition: infv∈ V(x,v)u(x)+f(x,u(x),∇ u(x) σ(x,v),v)\=0, x∈ D, and $u(x)=g(x)$, x∈ ∂ D, where D is a bounded set in Rᵈ, V is a compact set in Rᵏ, and for u∈ C²(D) and (x,v)∈ D× V, L(x,v)u(x):=1/2∑i,j=1ᵈ(σσ^)i,j(x,v) ∂²u∂ xᵢ∂ xⱼ(x) +∑ᵢ₌₁ᵈbᵢ(x,v)∂ u∂ xᵢ(x).