Paavo Salminen: Optimal stopping of strong Markov processes

The theory of optimal stopping is the crucial tool in, e.g.,

• sequential statistical testing of hypotheses,
• pricing of American options.

This survey talk is on methods for solving infinite horizon optimal stopping problems for continuous time strong Markov processes.  Given  a non-negative smooth reward function G the problem is to find a stopping time τ*

sup_{τ ∊ ?} ?_x (G(X_τ)) = ?_x (G(X_τ*)),

where X is the underlying process and ? is the set of all stopping times in the natural filtration of X.

We focus on verification theorems obtained by 

• principle of smooth pasting,
• Riesz representation for excessive functions,
• representing excessive functions as expected suprema.

Some examples are presented, in particular, for Lévy processes.

The talk is concluded with a short discussion on the historical development of the theory of optimal stopping.