Target-Generated Dirichlet Problems in State-Constrained Stochastic Control

Abstract

We study state-constrained stochastic control problems with a scalar surplus $R \geq 0$. Controls for which the drift and volatility of $R$ vanish at zero define a lower-dimensional Hamilton-Jacobi-Bellman equation. Starting from control-wise state-constraint viscosity inequalities, we prove that the upper and lower limits of the interior value are a subsolution and a supersolution of this boundary equation. Terminal compatibility and comparison identify their common limit. We require one-sided bounds on positive surplus drift and generator growth, together with domination by boundary generators. For compact or coercive controls, these follow from a local lower bound and continuity of the boundary control set. For a smooth stochastic target value $w$, the transformation $R = Y - w(t,X)$ flattens the viable epigraph, and, under the stated hypotheses, the boundary control value supplies the Dirichlet datum. Applications include set-valued boundary controls, unbounded drift with quadratic costs, and redundant hedging instruments with a nonconstant target boundary.