A sequential stopping problem with costly reversibility

Abstract We study sequential optimal stopping with partial reversibility. The optimal stopping problem is subject to implementation delay, which is random and exponentially distributed. Once the stopping decision is made, the decision maker can, by incurring a cost, call the decision off and restart the stopping problem. The optimization criterion is to maximize the expected present value of the total payoff. We characterize the value function in terms of a Bellman principle for a wide class of payoff functions and potentially multidimensional strong Markov dynamics. We also analyse the case of linear diffusion dynamics and characterize the value function and the optimal decision rule for a wide class of payoff functions.