Approximation of Infinitely Divisible Random Variables with Application to the Simulation of Stochastic Processes
(2001) Abstract
 This thesis consists of four papers A, B, C and D. Paper A and B treats the simulation of stochastic differential equations (SDEs). The research presented therein was triggered by the fact that there were not any efficient implementations of the higher order methods for simulating SDEs. So in practice the higher order methods required at least the same amount of work as the Euler method to obtain a given mean square error. The faster convergence rate of the higher order methods requires the simulation of the so called iterated Itô integrals. In (A) we use a shotnoise type series representation of one iterated Itô integral. We split the series representation into a sum of n terms and a remainder term and show that the remainder term is... (More)
 This thesis consists of four papers A, B, C and D. Paper A and B treats the simulation of stochastic differential equations (SDEs). The research presented therein was triggered by the fact that there were not any efficient implementations of the higher order methods for simulating SDEs. So in practice the higher order methods required at least the same amount of work as the Euler method to obtain a given mean square error. The faster convergence rate of the higher order methods requires the simulation of the so called iterated Itô integrals. In (A) we use a shotnoise type series representation of one iterated Itô integral. We split the series representation into a sum of n terms and a remainder term and show that the remainder term is asymptotically Gaussian as n goes to infinity. We provide an explicit coupling of the remainder a Gaussian random variable and show that this improves the mean square error by a factor n^½. In (B) we provide a multidimensional extension of the results in (A) as well as the not previously known simultaneous characteristic function of all iterated Itô integrals obtained the n pairing m independent Wiener processes. In (C) we study the simulation of type G Lévy processes. Recall that random variable is said to be of type G if it is a Gaussian variance mixture. We note that type G Lévy processes are subordinated Wiener processes. We use a series representation of the subordinator, a tailsum approximation and obtain an explicit coupling between type G Lévy processes and the sum of a compound Poisson process and a scaled Wiener process. We calculate the mean integrated square error for this approximation. We examine the convergence of the scaled tailsum process to its mean value function and provide a sufficient condition for this convergence. In paper (D) we utilise the coupling results from paper (C) to obtain approximations of stochastic integrals with respect to type G Lévy processes. Depending on the properties of the integrator we obtain either pointwise mean square error results or mean integrated square error results for the approximation. We also show that a stochastic time change representation of stochastic integrals can be used to obtain useful approximations. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/41348
 author
 Wiktorsson, Magnus ^{LU}
 supervisor

 Tobias Rydén ^{LU}
 opponent

 Professor Talay, Denis, INRIA, Sophia Antipolis (France)
 organization
 publishing date
 2001
 type
 Thesis
 publication status
 published
 subject
 keywords
 operations research, Statistics, aktuariematematik, Stochastic differential equation, Infinitely divisible distribution, Type G distribution, Lévy process, Stochastic integral, Mathematical Statistics, Matematik, Mathematics, programming, actuarial mathematics, Statistik, operationsanalys, programmering
 pages
 114 pages
 publisher
 Centre for Mathematical Sciences, Lund University
 defense location
 MH:C
 defense date
 20010323 10:15
 external identifiers

 other:ISRN: LUTFMS10142001
 ISBN
 916284640X
 language
 English
 LU publication?
 yes
 id
 44ba268fdf7b44cd9e2716763097cb4c (old id 41348)
 date added to LUP
 20070927 16:19:36
 date last changed
 20160919 08:45:06
@phdthesis{44ba268fdf7b44cd9e2716763097cb4c, abstract = {This thesis consists of four papers A, B, C and D. Paper A and B treats the simulation of stochastic differential equations (SDEs). The research presented therein was triggered by the fact that there were not any efficient implementations of the higher order methods for simulating SDEs. So in practice the higher order methods required at least the same amount of work as the Euler method to obtain a given mean square error. The faster convergence rate of the higher order methods requires the simulation of the so called iterated Itô integrals. In (A) we use a shotnoise type series representation of one iterated Itô integral. We split the series representation into a sum of n terms and a remainder term and show that the remainder term is asymptotically Gaussian as n goes to infinity. We provide an explicit coupling of the remainder a Gaussian random variable and show that this improves the mean square error by a factor n^½. In (B) we provide a multidimensional extension of the results in (A) as well as the not previously known simultaneous characteristic function of all iterated Itô integrals obtained the n pairing m independent Wiener processes. In (C) we study the simulation of type G Lévy processes. Recall that random variable is said to be of type G if it is a Gaussian variance mixture. We note that type G Lévy processes are subordinated Wiener processes. We use a series representation of the subordinator, a tailsum approximation and obtain an explicit coupling between type G Lévy processes and the sum of a compound Poisson process and a scaled Wiener process. We calculate the mean integrated square error for this approximation. We examine the convergence of the scaled tailsum process to its mean value function and provide a sufficient condition for this convergence. In paper (D) we utilise the coupling results from paper (C) to obtain approximations of stochastic integrals with respect to type G Lévy processes. Depending on the properties of the integrator we obtain either pointwise mean square error results or mean integrated square error results for the approximation. We also show that a stochastic time change representation of stochastic integrals can be used to obtain useful approximations.}, author = {Wiktorsson, Magnus}, isbn = {916284640X}, keyword = {operations research,Statistics,aktuariematematik,Stochastic differential equation,Infinitely divisible distribution,Type G distribution,Lévy process,Stochastic integral,Mathematical Statistics,Matematik,Mathematics,programming,actuarial mathematics,Statistik,operationsanalys,programmering}, language = {eng}, pages = {114}, publisher = {Centre for Mathematical Sciences, Lund University}, school = {Lund University}, title = {Approximation of Infinitely Divisible Random Variables with Application to the Simulation of Stochastic Processes}, year = {2001}, }