Some Markov Processes in Finance and Kinetics
This thesis consists of four papers. The first two papers treat extremes for L\'evy processes, while papers three and four treat the Kac model with unbounded collision kernel.
The L\'evy process papers relate the distribution of the supremum of a L\'evy process over a compact time interval to the distribution of the process value at the right endpoint of this interval. L\'evy processes are sorted into different classes depending on the tails of their univariate marginal distributions. In the first paper we treat processes with heavier tails, while processes with lighter tails are handled in the second paper. Our results are applicable to many processes recently introduced in mathematical finance. For instance, they may be used to approximate the distribution of the maximum of a stock price over a finite time span.
The papers on the Kac model mainly deal with an approximation of the Kac model with unbounded collsion kernel where small jumps are replaced by a Brownian motion. In the first and more theoretical of these papers we prove convergence of the approximating processes to the process with unbounded collision kernel. We also give results on the spectral gap of the Kac model with unbounded collision kernel. In the second paper on the Kac model we present numerical results which show that our approximation scheme gives a considerable improvement of the standard approximation which uses only a truncated collision kernel and that this improvement is more obvious as the collsion kernel gets more singular. Our numerical investigations are carried out for the Kac model with Gaussian thermostat as well as for a more physically relevant three-dimensional model.
Extreme value theory
Semi-heavy tailed distirbution
Generalized hyperbolic process
Direct simulation Monte Carlo