Nonexistence of nonconstant solutions of some degenerate Bellman equations and applications to stochastic control

TL;DR

This paper proves that certain degenerate Bellman equations in bounded domains admit only constant solutions under boundary growth conditions, with applications to stochastic control problems like exit times and ergodic control with constraints.

Contribution

It establishes a nonexistence result for nonconstant solutions of degenerate Bellman equations and applies this to key stochastic control problems.

Findings

Sub- and supersolutions with controlled boundary growth are constant.

The boundary degeneracy and drift conditions ensure domain invariance for controlled diffusions.

Results apply to exit problems and small discount limits in ergodic control.

Abstract

For a class of Bellman equations in bounded domains we prove that sub- and supersolutions whose growth at the boundary is suitably controlled must be constant. The ellipticity of the operator is assumed to degenerate at the boundary and a condition involving also the drift is further imposed. We apply this result to stochastic control problems, in particular to an exit problem and to the small discount limit related with ergodic control with state constraints. In this context, our condition on the behavior of the operator near the boundary ensures some invariance property of the domain for the associated controlled diffusion process.