Global regularity of the value function in a stopper vs. singular-controller game

TL;DR

This paper proves the smoothness of the value function in a complex stochastic game involving stopping and singular control, revealing its regularity and boundary behavior, which aids in understanding optimal strategies.

Contribution

It establishes the $C^1$ regularity of the value function and characterizes the continuity of second derivatives, advancing the analysis of free-boundaries in stochastic games.

Findings

Value function is $C^1$ in the domain.

Second derivatives are continuous except across the stopping boundary.

Discontinuity in derivatives occurs naturally at the free boundary.

Abstract

We study a class of zero-sum stochastic games between a stopper and a singular-controller, previously considered in [Bovo and De Angelis (2025)]. The underlying singularly-controlled dynamics takes values in $\mathcal{O}\subseteq\mathbb{R}$. The problem is set on a finite time-horizon and is connected to a parabolic variational inequality of min-max type with spatial-derivative and obstacle constraints. We show that the value function of the problem is of class $C^1$ in the whole domain $[0,T)\times\mathcal{O}$ and that the second-order spatial derivative and the second-order mixed derivative are continuous everywhere except for a (potential) jump across a non-decreasing curve (the stopping boundary of the game). The latter discontinuity is a natural consequence of the partial differential equation associated to the problem. Beyond its intrinsic analytical value, such a regularity for the value function is a stepping stone for further exploring the structure and properties of the free-boundaries of the stochastic game, which in turn determine the optimal strategies of the players.