Fluctuation theory and stochastic games for spectrally negative Lévy processes

Abstract

Lévy processes have stationary, independent increments. This seemingly unassuming (defining) property leads to a surprisingly rich class of processes which appear in a large number of applications including queueing, fragmentation theory, branching processes, dams, risk theory and finance. In this thesis we study various aspects and applications of Lévy processes. We find the Laplace transform of the last time before an exponential time that a spectrally negative process is negative. This result is inspired by the extension of the model considered by Lundberg: instead of a compound Poisson process (with negative jumps and positive drift) the risk process is modelled by a general spectrally negative Lévy process. The spectral negativity of the process stems from the fact that jumps in the risk process are caused by the incoming claims and hence only occur in the negative direction. Scale functions play an important role in the results. We use excursion theory to express resolvent measure of reflected Lévy processes in terms of the resolvent measure of the (unreflected) process. We apply this result to find explicitly the potential measure of a reflected symmetric stable process killed at exceeding a certain level. We also study optimal stopping problems for generalised Ornstein--Uhlenbeck processes driven by a spectrally negative Lévy process. We find the optimal stopping time as well as sufficient conditions for smooth fit to hold. The rest of this thesis is devoted to stochastic games for spectrally negative Lévy processes. Such a game can be interpreted as a two player version of an optimal stopping problem. We study two specific examples: an extension of the American put option (called the McKean game) and an extension of the Russian option (Shepp--Shiryaev game). We use a combination of martingale techniques, fluctuation theory of spectrally negative Lévy processes and auxiliary optimal stopping problems to find the optimal stopping strategies for both players. We find that the jumps of the process have an important influence on the optimal strategies: in some sense one could argue that the players becomes more defensive, as the jumps of the process could lead to sudden changes in value of the underlying process. We show that smooth fit holds at a boundary point of the stopping region according to whether that point is regular for the interior of the stopping region.