The eigenvalue problem of singular ergodic control

TL;DR

This paper investigates a PDE related to ergodic control, establishing the existence and uniqueness of a key eigenvalue and solution, with implications for long-term average costs in singular control problems.

Contribution

It proves the existence and uniqueness of the eigenvalue and viscosity solution for a specific PDE, providing min-max formulas and a probabilistic interpretation.

Findings

Unique eigenvalue lambda* exists with a corresponding viscosity solution u*

Lambda* has a probabilistic interpretation as the ergodic cost in control problems

Provides min-max formulas for lambda*

Abstract

We consider the problem of finding a real number lambda and a function u satisfying the PDE max{lambda -\Delta u -f,|Du|-1}=0, for all x in R^n. Here f is a convex, superlinear function. We prove that there is a unique lambda* such that the above PDE has a viscosity solution u satisfying u(x)/|x|->1 as |x| tends to infinity. Moreover, we show that associated to lambda^* is a convex solution u^* with D^2u^* uniformly bounded and give two min-max formulae for lambda^*. lambda^* has a probabilistic interpretation as being the least, long-time averaged ("ergodic") cost for a singular control problem involving f.