Optimal stopping problems for the maximum process

A cornerstone in the theory of optimal stopping for the maximum process is a result known as Peskir's maximality principle. It has proved to be a powerful tool to solve optimal stopping problems involving the maximum process under the assumption that the driving process X is a time-homogeneous diffusion. In this thesis we adapt Peskir's maximality principle to allow for X a spectrally negative Levy processes, thereby providing a general method to approach optimal stopping problems for the maximum process driven by spectrally negative Levy processes. We showcase this by explicitly solving three optimal stopping problems and the capped versions thereof. Here capped version means a modification of the original optimal stopping problem in the sense that the payoff is bounded from above by some constant. Moreover, we discuss applications of the aforementioned optimal stopping problems in option pricing in financial markets whose price process is driven by an exponential spectrally negative Levy process. Finally, to further highlight the applicability of our general method, we present the solution to the problem of predicting the time at which a positive self-similar Markov process with one-sided jumps attains its maximum or minimumEThOS - Electronic Theses Online ServiceGBUnited Kingdo