Variational inequalities on unbounded domains for zero-sum singular-controller vs. stopper games

TL;DR

This paper analyzes zero-sum games involving a singular-controller and a stopper within unbounded domains, establishing the existence of a game value and optimal strategies through variational inequalities without boundedness restrictions.

Contribution

It proves the existence of a game value and characterizes it as a maximal Sobolev solution of a min-max variational inequality on unbounded domains, without boundedness assumptions.

Findings

Existence of a game value for the singular-controller vs. stopper game.

Characterization of the value as a maximal Sobolev solution.

Derivation of optimal strategies for the stopper.

Abstract

We study a class of zero-sum games between a singular-controller and a stopper over finite-time horizon. The underlying process is a multi-dimensional (locally non-degenerate) controlled stochastic differential equation (SDE) evolving in an unbounded domain. We prove that such games admit a value and provide an optimal strategy for the stopper. The value of the game is shown to be the maximal solution, in a suitable Sobolev class, of a variational inequality of `min-max' type with obstacle constraint and gradient constraint. Although the variational inequality and the game are solved on an unbounded domain we do not require boundedness of either the coefficients of the controlled SDE or of the cost functions in the game.