Zero-Sum Dynkin Games under Common and Independent Poisson Constraints

Abstract.

Zero-sum Dynkin games under Poisson constraints, where players can only stop at the event times of a Poisson process, have been studied widely in the recent literature. The constraint can be modeled in two ways: either both players share the same Poisson process (the common constraint) or each player has their own Poisson process (the independent constraint). In a Markovian framework, where payoffs are functions of an underlying diffusion, we establish necessary and sufficient conditions for the equivalence of the game’s solution—comprising the value function and optimal stopping sets—under the common and independent constraints. Specifically, if the stopping sets of the maximizer and minimizer in the game under the common constraint are disjoint, then the solution to the game is the same under both the common and the independent constraint. However, the fact that the stopping sets are disjoint in the game under the independent constraint, is not sufficient to guarantee that the solution of the game under the independent constraint is also the solution under the common constraint. To demonstrate the broad applicability of our results, we solve infinite-horizon Dynkin games satisfying the assumptions of our main theorems, using backward stochastic differential equation (BSDE) techniques. This requires extending standard BSDE results from the finite-horizon setting to the infinite-horizon case, allowing for unbounded solutions.