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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Andrea Bovo & Tiziano De Angelis & Elena Issoglio, 2025. "Variational Inequalities on Unbounded Domains for Zero-Sum Singular Controller vs. Stopper Games," Mathematics of Operations Research, INFORMS, vol. 50(1), pages 277-312, February.
    • Handle: RePEc:inm:ormoor:v:50:y:2025:i:1:p:277-312
    DOI: 10.1287/moor.2023.0029
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