Optimal Stopping with Random Intervention Times

We consider a class of optimal stopping problems where the ability to stop depends on exogenous Poisson signal process – one can only stop at the Poisson jump times. Even though the time variable in these problems has a discrete aspect, a variational inequality can be obtained by considering an underlying continuous time structure. Depending on whether stopping is allowed at t = 0, the value function exhibits different properties across the optimal exercise boundary. Indeed, the value function is only C0 across the optimal boundary when stopping is allowed at t = 0 and C2 otherwise, both contradicting the usual C1 smoothness that is necessary and sufficient for the application of the principle of smooth fit. Also discussed is an equivalent stochastic control formulation for these stopping problems. Finally, we derive the asymptotic behavior of the value functions and optimal exercise boundaries as the intensity of the Poisson process goes to infinity, or roughly speaking, as the problems converge to the classical continuous-time optimal stopping problems.