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

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 model of proof is based on reducting 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 ; It^o's formula