Zero-sum stopper vs. singular-controller games with constrained control directions

TL;DR

This paper studies a class of zero-sum games involving a stopper and a singular controller with restricted control directions, establishing the existence of the game's value and optimal strategies through an approximation approach.

Contribution

It introduces an approximation method for zero-sum stopper vs. singular-controller games with directional control constraints, proving value existence and optimal strategies.

Findings

Proved existence of the game's value.

Established convergence of approximation procedures.

Derived optimal strategies for the stopper.

Abstract

We consider a class of zero-sum stopper vs. singular-controller games in which the controller can only act on a subset $d_0<d$ of the $d$ coordinates of a controlled diffusion. Due to the constraint on the control directions these games fall outside the framework of recently studied variational methods. In this paper we develop an approximation procedure, based on $L^1$-stability estimates for the controlled diffusion process and almost sure convergence of suitable stopping times. That allows us to prove existence of the game's value and to obtain an optimal strategy for the stopper, under continuity and growth conditions on the payoff functions. This class of games is a natural extension of (single-agent) singular control problems, studied in the literature, with similar constraints on the admissible controls.