Ruin probabilities and first passage times for selfsimilar processes.
Research output: Thesis › Doctoral Thesis (compilation)
Abstract
This thesis investigates ruin probabilities and first passage times for selfsimilar processes.
We propose selfsimilar 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.
We propose selfsimilar 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
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Research areas and keywords  Subject classification (UKÄ) – MANDATORY
Keywords

Original language  English 

Qualification  Doctor 
Awarding Institution  
Supervisors/Assistant supervisor 

Award date  1998 Nov 6 
Publisher 

Print ISBNs  9162831666 
Publication status  Published  1998 
Publication category  Research 
Bibliographic note
Defence details
Date: 19981106
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
