A zero-sum game between a singular stochastic controller and a discretionary stopper

TL;DR

This paper analyzes a zero-sum stochastic differential game involving a singular controller and a discretionary stopper, deriving variational inequalities and explicit solutions for special cases, highlighting the non-smoothness of value functions.

Contribution

It introduces a new framework for zero-sum games with singular control and stopping, providing variational inequalities and explicit solutions, and exploring the non-smoothness and strategy non-uniqueness.

Findings

Explicit solutions for two special cases.

Value functions generally lack $C^1$ smoothness.

Non-uniqueness of optimal strategies when the controller moves first.

Abstract

We consider a stochastic differential equation that is controlled by means of an additive finite-variation process. A singular stochastic controller, who is a minimizer, determines this finite-variation process, while a discretionary stopper, who is a maximizer, chooses a stopping time at which the game terminates. We consider two closely related games that are differentiated by whether the controller or the stopper has a first-move advantage. The games' performance indices involve a running payoff as well as a terminal payoff and penalize control effort expenditure. We derive a set of variational inequalities that can fully characterize the games' value functions as well as yield Markovian optimal strategies. In particular, we derive the explicit solutions to two special cases and we show that, in general, the games' value functions fail to be $C^1$. The nonuniqueness of the optimal strategy is an interesting feature of the game in which the controller has the first-move advantage.