[Paper Review] Variational inequalities on unbounded domains for zero-sum singular-controller vs. stopper games

Andrea Bovo, Tiziano De Angelis+1 more|arXiv (Cornell University)|Mar 11, 2022

Stochastic processes and financial applications4 citations

TL;DR

This paper establishes the existence of a value and optimal strategies for zero-sum stochastic games between a singular controller and a stopper on unbounded domains, using variational inequalities with min-max structure, obstacle constraints, and gradient constraints. The key contribution is proving the value is the maximal solution in a Sobolev class to a parabolic variational inequality without requiring boundedness of coefficients or uniform ellipticity.

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.

Motivation & Objective

• To establish the existence of a value for zero-sum stochastic games involving singular control and optimal stopping on unbounded domains.
• To characterize the value function as the maximal solution in a Sobolev space to a variational inequality with min-max structure.
• To resolve the challenge of solving variational inequalities with two hard constraints—obstacle and gradient—on unbounded domains.
• To develop a unified analytical and probabilistic framework that does not require boundedness of SDE coefficients or payoff functions.
• To provide a foundation for future work on free-boundary problems and saddle-point analysis in controller-stopper games.

Proposed method

• Formulate the game as a parabolic variational inequality of 'min-max' type with obstacle constraint $ u \geq g $ and gradient constraint $ |\nabla u|_d \leq f $.
• Use penalization techniques to simultaneously address the obstacle and gradient constraints in the variational inequality.
• Establish uniform bounds on the Sobolev norm of solutions to the penalized PDEs using analytical tools from Evans and new probabilistic estimates.
• Construct a sequence of approximating solutions on bounded domains and extend them to the unbounded domain via localization and stopping time arguments.
• Prove existence and uniqueness of solutions to the penalized problem on both bounded and unbounded domains using fixed-point arguments and compactness.
• Apply Shaefer’s fixed point theorem to establish existence of a solution to the fully coupled variational inequality system.

Experimental results

Research questions

• RQ1 Does a zero-sum stochastic game between a singular controller and a stopper on an unbounded domain admit a value function?
• RQ2 Can the value function be characterized as the maximal solution in a Sobolev space to a variational inequality with min-max structure and two hard constraints?
• RQ3 Is it possible to solve such variational inequalities without assuming boundedness of the SDE coefficients or payoff functions?
• RQ4 Can the solution be constructed via penalization methods that simultaneously handle obstacle and gradient constraints?
• RQ5 Does the optimal stopping rule for the stopper admit a free-boundary representation that enables reformulation as a singular control problem?

Key findings

• The game admits a value, which is the maximal solution in a suitable Sobolev class to the variational inequality (1.4) with min-max structure.
• The value function satisfies the variational inequality a.e. in $[0,T) \times \mathbb{R}^d$ with terminal condition $ u(T,x) = g(T,x) $.
• The solution is constructed via a penalization method that simultaneously handles the obstacle constraint $ u \geq g $ and the gradient constraint $ |\nabla u|_d \leq f $.
• Uniform bounds on the Sobolev norm of the penalized solutions are established without requiring boundedness of coefficients or uniform ellipticity.
• An optimal stopping time $ \tau_* $ is explicitly characterized, enabling reformulation of the game as a singular control problem with absorption on the contact set $ \{v = g\} $.
• The existence and uniqueness of solutions to the penalized PDEs are rigorously proven for both bounded and unbounded domains, providing a foundation for future analysis.