Ruin Probabilities and First Passage Times for Self-Similar Processes

Zbigniew Michna

Department of Mathematical Statistics,
Lund Institute of Technology,
Lund University,
1998

ISBN 91-628-3166-6
ISRN LUNFD6/NFMS--1007--SE


Abstract:
This thesis  investigates ruin probabilities and first passage times for self-similar processes.

We propose self-similar processes as a risk model with claims appearing in good and bad periods. Then, in particular, we get the fractional Brownian motion with drift as a limit risk process. Some bounds and asymptotics for ruin probability on a finite interval for fractional Brownian motion are derived. A method of simulation of ruin probability over finite horizon for fractional Brownian motion is presented. The moments of the first passage time of fractional Brownian motion are studied. As an application of our method we numerically compute the Picands constant for fractional Brownian motion.

An asymptotic behavior of the supremum of a Gaussian process X over infinite horizon is studied. In particular, X can be fractional Brownian motion, a nonlinearly scaled Brownian motion or integrated stationary Gaussian processes.

The thesis treats first passage times and the expected number of crossings for symmetric alpha-stable processes. We derive Rice's formula for a class of stable processes and give a numerical approximation of the expected number of crossings based on Rice's formula.

We study weak convergence of a sequence of renewal processes constructed by a sequence of random variables belonging to the domain of attraction of a stable law. We show that this sequence is not tight in the Skorokhod topology but the weak convergence of some functionals is derived. A weaker notion of the weak convergence is proposed. As an application of our tools we present limit theorems for queues in heavy traffic.

Key words:
Ruin Probability, First Passage Time, Gaussian Process, Self-Similar Process, alpha-Stable Process, Lévy Motion, Renewal Process, Fractional Brownian Motion, Long Range Dependence, Scaled Brownian Motion, Risk Model, Fluid Model, Rice's Formula, Weak Convergence, Skorokhod Topology, Monte Carlo Method, Simulation of Ruin Probability, Exponential Bound, Picands Constant.