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

2025/06/23 by Bovo, Andrea, Milazzo, Alessandro
#35R35 #49N60 #60G40 #91A05 #91A15 #93E20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR)

paper · doi:10.48550/arxiv.2506.19129

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 O⊆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 C¹ in the whole domain [0,T)\timesO 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.

Citations

• Finite-time horizon, stopper vs. singular-controller games on the half-line
• On the saddle point of a zero-sum stopper vs. singular-controller game
• Stopper vs. singular-controller games with degenerate diffusions
• Zero-sum stopper vs. singular-controller games with constrained control directions
• Variational inequalities on unbounded domains for zero-sum\n singular-controller vs. stopper games
• How to detect a salami slicer: a stochastic controller-stopper game with unknown competition
• Density and gradient estimates for non degenerate Brownian SDEs with unbounded measurable drift
• A Dynkin game on assets with incomplete information on the return
• Reflected BSDEs with nonpositive jumps, and controller-and-stopper games
• Games of singular control and stopping driven by spectrally one-sided Levy processes
• On the Multi-Dimensional Controller and Stopper Games

Related

• On the saddle point of a zero-sum stopper vs. singular-controller game
• Variational inequalities on unbounded domains for zero-sum\n singular-controller vs. stopper games
• Finite-time horizon, stopper vs. singular-controller games on the half-line
• A geometric answer to an open question of singular control with stopping
• Global C¹ Regularity of the Value Function in Optimal Stopping\n Problems