Averaging problems of running processes associated with Brownian motion and applications

Recently the new technique to solve optimal stopping problems for Hunt processes is developed (see [S. Christensen, P. Salminen and B. Q. Ta, Optimal stopping of strong Markov processes, Stochastic Process. Appl. 123(3) (2013) 1138–1159]). The crucial feature of the approach is to utilize the representation of the r-excessive functions as expected suprema. However, it seems to be difficult when applying directly the approach to some concrete cases, e.g. one-sided problem for reflecting Brownian motion and two-sided problem for Brownian motion. In this paper, we review and exploit this approach to find explicit solutions of two problems above.