A Note on Optimal Stopping in Models with Delay

It is known that optimal stopping problems form an important class of optimal control problems having applications in stochastic calculus (maximal inequalities), statistics (sequential analysis) and mathematical finance (American options). The results about the relationship between optimal stopping problems for Markov processes and free-boundary problems for partial differential equations often give an opportunity to obtain explicit solutions in some particular cases (see e.g. Shiryaev [11] or [12]). In the present paper we consider some optimal stopping problems in a model with processes solving stochastic delay differential equations. In recent years several stochastic control problems for models described by stochastic delay differential equations were studied. Øksendal and Sulem [7] proved maximum principles for certain classes of such models and applied