Excursions of Levy processes and applications in mathematical finance and insurance

The excursion time of a Levy process measures the time it spends continuously below or above a given barrier. This thesis contains five papers dealing with the excursions of different Levy processes and their applications in mathematical finance and insurance. Each of the five papers is presented in one of the chapters of this thesis starting from Chapter 2. In Chapter 2 the excursions of a Brownian motion with drift below or above a given barrier are studied by using a two-state semi-Markov model. Based on the results single barrier two-sided Parisian options are studied and the explicit expressions for the Laplace transforms of their price formulae are given. In Chapter 3 the excursion time of a Brownian motion with drift outside a corridor is considered by using a four-state semi-Markov model. The results are used to obtain the explicit expressions for the Laplace transforms of the prices of the double barrier Parisian options. In Chapter 4 Parisian corridor options are introduced and priced by using the results of the excursion time of a Brownian motion with drift inside a corridor. In Chapter 5 the main focus is the excursions of a Levy process with negative exponential jumps below a given barrier. Based on the results, a Parisian option whose underlying asset price follows this process is priced, as well as a Parisian type digital option. This is the first ever attempt to price Parisian options involving jump processes. Furthermore, the concept of ruin in risk theory is extended to the Parisian type of ruin. In Chapter 6 the excursions of a risk surplus process with a more general claim distribution are considered. For the processes without initial reserve, the Parisian ruin probability in an infinite time horizon is calculated. For the positive initial reserve case, only the asymptotic form can be obtained for very large initial reserve and small claim distributions.