A saddle point in stopper vs. singular-controller games with free boundaries

Bovo Andrea, Università di Torino

In this work, we look for a saddle point for a class of Stopper vs. Singular-Controller games over a finite-time horizon. In this game the controller (the minimiser) can choose a control from the class of singular controls, whereas the stopper (the maximiser) can choose the time at which the game ends. We prove that the game admits a value function which is related to a variational inequality with two constraints: an obstacle constraint and a gradient constraint. These constraints lead to two continuous free-boundaries which divide the whole `space' in three regions. We prove that the value function is continuously differentiable in both time and space, and the mixed derivative and the second order spatial derivative are continuous across the free boundary associated to the controller. Finally, we obtain an optimal strategy for both the controller and the stopper player.

Area: CS6 - Stochastic optimal control, BSDEs, and applications (Fulvia Confortola and Alessandro Calvia)

Keywords: 2-person games, Stochastic games, optimal stochastic control, optimal stopping problems, free boundary problems for PDEs

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