An optimal stopping problem in a diffusion-type model with delay

Abstract

We present an explicit solution to an optimal stopping problem in a model described by a stochastic delay differential equation with an exponential delay measure. The method of proof is based on reducing the initial problem to a free-boundary problem and solving the latter by means of the smooth-fit condition. The problem can be interpreted as pricing special perpetual average American put options in a diffusion-type model with delay.Optimal stopping, stochastic delay differential equation, diffusion process, sufficient statistic, free-boundary problem, smooth fit, Girsanov’s theorem, Ito’s formula