Variational Inequalities on Unbounded Domains for Zero-Sum Singular Controller vs. Stopper Games

Andrea Bovo (), Tiziano De Angelis () and Elena Issoglio ()
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Andrea Bovo: Department of Economics, Social Studies, Applied Mathematics and Statistics, School of Management and Economics, University of Torino, 10134 Torino, Italy
Tiziano De Angelis: Department of Economics, Social Studies, Applied Mathematics and Statistics, School of Management and Economics, University of Torino, 10134 Torino, Italy; and Collegio Carlo Alberto, 10122 Torino, Italy
Elena Issoglio: Department of Mathematics “G. Peano,” University of Torino, 10123 Torino, Italy

Mathematics of Operations Research, 2025, vol. 50, issue 1, 277-312

Abstract: We study a class of zero-sum games between a singular controller and a stopper over a finite-time horizon. The underlying process is a multidimensional (locally nondegenerate) 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 an obstacle constraint and a 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.

Keywords: Primary 91A15; 49J40; zero-sum stochastic games; singular control; optimal stopping; controlled diffusions; variational inequalities; obstacle problems; gradient constraints; penalization methods; unbounded domains (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:50:y:2025:i:1:p:277-312

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