Ruin probabilities and first passage times for selfsimilar processes.
(1998) 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... (More)  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. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/39017
 author
 Michna, Zbigniew ^{LU}
 supervisor
 opponent

 Dr Norros, Ilkka, VTT Information Technology, P.O. Box 1202, 02044 VTT, Finland
 organization
 publishing date
 1998
 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, SelfSimilar 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 hÃ¶rsal B
 defense date
 19981106 10:15:00
 external identifiers

 other:ISRN: LUNFD6/NFMS1007SE
 ISBN
 9162831666
 language
 English
 LU publication?
 yes
 id
 5191f1c83d0b4b99b10feab32713aca7 (old id 39017)
 date added to LUP
 20160404 11:06:33
 date last changed
 20181121 21:02:42
@phdthesis{5191f1c83d0b4b99b10feab32713aca7, abstract = {{This thesis investigates ruin probabilities and first passage times for selfsimilar processes.<br/><br> <br/><br> 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.<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 = {{9162831666}}, 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; SelfSimilar Process; Gaussian Process; Ruin Probability; First Passage Time; Exponential Bound; Picands Constant.; Mathematics; Matematik}}, language = {{eng}}, publisher = {{Printed in Sweden by KFS AB, LUND 1998}}, school = {{Lund University}}, title = {{Ruin probabilities and first passage times for selfsimilar processes.}}, year = {{1998}}, }