Excessive functions, appell polynomials and optimal stopping

The main topic of the thesis is optimal stopping. This is treatedin two research articles. In the first article we introduce a new approachto optimal stopping of general strong Markov processes. Theapproach is based on the representation of excessive functions as expectedsuprema. We present a variety of examples, in particular, theNovikov-Shiryaev problem for Lévy processes. In the second articleon optimal stopping we focus on differentiability of excessive functionsof diffusions and apply these results to study the validity of theprinciple of smooth fit. As an example we discuss optimal stopping ofsticky Brownian motion. The third research article offers a survey likediscussion on Appell polynomials. The crucial role of Appell polynomialsin optimal stopping of Lévy processes was noticed by Novikovand Shiryaev. They described the optimal rule in a large class of problemsvia these polynomials. We exploit the probabilistic approach toAppell polynomials and show that many classical results are obtainedwith ease in this framework. In the fourth article we derive a newrelationship between the generalized Bernoulli polynomials and thegeneralized Euler polynomials.