Optimal stopping with variable attention

Abstract

We consider an optimal stopping problem of a linear diffusion under Poisson constraint where the agent can adjust the arrival rate of new stopping opportunities. We assume that the agent may switch the rate of the Poisson process between two values. Maintaining the lower rate incurs no cost, whereas the higher rate requires effort that is captured by a cost function c. We study a broad class of payoff functions, cost functions and diffusion dynamics, for which we explicitly characterize the solution to the constrained stopping problem. We also characterize the case where switching to the higher rate is always suboptimal. The results are illustrated with two examples.