Ruin probabilities and first passage times for self-similar processes.

Research output: ThesisDoctoral Thesis (compilation)

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 infinite 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 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.

Details

Authors
  • Zbigniew Michna
Organisations
Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Probability Theory and Statistics

Keywords

  • Simulation of Ruin Probability, Monte Carlo Method, Skorokhod Topology, Weak Convergence, Rice's Formula, Fluid Model, Risk Model, Scaled Brownian Motion, Long Range Dependence, Fractional Brownian Motion, Renewal Process, Levy Motion, Stable Process, Self-Similar Process, Gaussian Process, Ruin Probability, First Passage Time, Exponential Bound, Picands Constant., Mathematics, Matematik
Original languageEnglish
QualificationDoctor
Awarding Institution
Supervisors/Assistant supervisor
  • [unknown], [unknown], Supervisor, External person
Award date1998 Nov 6
Publisher
  • Printed in Sweden by KFS AB, LUND 1998
Print ISBNs91-628-3166-6
Publication statusPublished - 1998
Publication categoryResearch

Bibliographic note

Defence details Date: 1998-11-06 Time: 10:15 Place: Mattehusets hörsal B External reviewer(s) Name: Norros, Ilkka Title: Dr Affiliation: VTT Information Technology, P.O. Box 1202, 02044 VTT, Finland ---