Nonzero-Sum Optimal Stopping Game with Continuous vs. Periodic Exercise Opportunities

We introduce a new nonzero-sum game of optimal stopping with asymmetric exercise opportunities. Given a stochastic process modeling the value of an asset, one player observes and can act on the process continuously, whereas the other player can act on it only periodically at independent Poisson arrival times. The first one to stop receives a reward, different for each player, whereas the other one gets nothing. We study how each player balances the maximization of gains against the maximization of the likelihood of stopping before the opponent. In such a setup driven by a Le<acute accent>vy <acute accent>vy process with positive jumps, we not only prove the existence but also explicitly construct a Nash equilibrium with values of the game written in terms of the scale function. Numerical illustrations with put-option payoffs are also provided to study the behavior of the players' strategies as well as the quantification of the value of available exercise opportunities.