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

Michna, Zbigniew LU (1998)
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... (More)
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. (Less)
Please use this url to cite or link to this publication:
author
opponent
  • Dr Norros, Ilkka, VTT Information Technology, P.O. Box 1202, 02044 VTT, Finland
organization
publishing date
type
Thesis
publication status
published
subject
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
pages
117 pages
publisher
Printed in Sweden by KFS AB, LUND 1998
defense location
Mattehusets hörsal B
defense date
1998-11-06 10:15
external identifiers
  • Other:ISRN: LUNFD6/NFMS--1007--SE
ISBN
91-628-3166-6
language
English
LU publication?
yes
id
5191f1c8-3d0b-4b99-b10f-eab32713aca7 (old id 39017)
date added to LUP
2007-08-01 14:52:09
date last changed
2016-09-19 08:45:08
@misc{5191f1c8-3d0b-4b99-b10f-eab32713aca7,
  abstract     = {This thesis investigates ruin probabilities and first passage times for self-similar processes.<br/><br>
<br/><br>
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.<br/><br>
<br/><br>
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.<br/><br>
<br/><br>
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.<br/><br>
<br/><br>
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.},
  author       = {Michna, Zbigniew},
  isbn         = {91-628-3166-6},
  keyword      = {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},
  language     = {eng},
  pages        = {117},
  publisher    = {ARRAY(0x9d33660)},
  title        = {Ruin probabilities and first passage times for self-similar processes.},
  year         = {1998},
}