Stopper vs. Singular Controller Games With Degenerate Diffusions

Abstract

We study zero-sum stochastic games between a singular controller and a stopper when the (state-dependent) diffusion matrix of the underlying controlled diffusion process is degenerate. In particular, we show the existence of a value for the game and determine an optimal strategy for the stopper. The degeneracy of the dynamics prevents the use of analytical methods based on solution in Sobolev spaces of suitable variational problems. Therefore we adopt a probabilistic approach based on a perturbation of the underlying diffusion modulated by a parameter \(\gamma >0\). For each \(\gamma >0\) the approximating game is non-degenerate and admits a value \(u^\gamma \) and an optimal strategy \(\tau ^\gamma _*\) for the stopper. Letting \(\gamma \rightarrow 0\) we prove convergence of \(u^\gamma \) to a function v, which identifies the value of the original game. We also construct explicitly optimal stopping times \(\theta ^\gamma _*\) for \(u^\gamma \), related but not equal to \(\tau ^\gamma _*\), which converge almost surely to an optimal stopping time \(\theta _*\) for the game with degenerate dynamics.