Topics in Simulation and Stochastic Analysis
(2003) Abstract
 Paper A investigates how to simulate a differentiated mean in cases where interchanging differentiation and expectation is not allowed. Three approaches are available, finite differences (FD's), infinitesimal perturbation analysis (IPA) and the likelihood ratio score function (LRSF) method. We study FD's under discontinuities and show that the optimal decay rate of the mean square error is typically like n^{4/5}. The IPA method is generalized to allow for random variables with a finite number of jumps. Finally, we give a unified view of IPA and LRSF, which shows that, in the setting we consider, they are actually identical as long as the mathematics goes.
Paper B provides error rates related to simulation of a Levy... (More)  Paper A investigates how to simulate a differentiated mean in cases where interchanging differentiation and expectation is not allowed. Three approaches are available, finite differences (FD's), infinitesimal perturbation analysis (IPA) and the likelihood ratio score function (LRSF) method. We study FD's under discontinuities and show that the optimal decay rate of the mean square error is typically like n^{4/5}. The IPA method is generalized to allow for random variables with a finite number of jumps. Finally, we give a unified view of IPA and LRSF, which shows that, in the setting we consider, they are actually identical as long as the mathematics goes.
Paper B provides error rates related to simulation of a Levy process when small jumps have been truncated. Three different truncations schemes are considered. Question: How to chose the truncation threshold? Answer: Weak error rates. We also consider a process X of infinite variation and the finest approximation in more detail and give a general Edgeworth expansion for a triangular array of Levy processes with third Levy moment 0.
Paper C treats the inhomogeneous stochastic Cauchy problem for the wave equation. The initial values and driving force are assumed to be given stochastic distribution valued functions. We show existence and uniqueness and the solution $U(t,x)=U(t,x,omega)$ satisfies the equation in the strong sense with respect to time and space parameters $t$ and $x$.
Paper D deals with a class of nonlinear heat equations driven by the spacetime derivative of a Brownian sheet. An Eulerlike approximation scheme is applied in order to solve the equation numerically. We show that the approximation scheme converges uniformly in probability.
Finally, the note on a paperinprogres in chapter E deals with a digital communication system considered in a setting where the bit error probabilitiy p is so small that crude Monte Carlo simulation is not feasible for evaluating p and further related characteristics. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/466076
 author
 Signahl, Mikael ^{LU}
 opponent

 Professor Kaj, Ingemar, Uppsala
 organization
 publishing date
 2003
 type
 Thesis
 publication status
 published
 subject
 keywords
 Mathematics, Matematik, stochastic heat equation, stochastic wave equation, error rates, truncated Levy process, discontinuities, optimal decay rate, digital communication system
 pages
 137 pages
 publisher
 Centre for Mathematical Sciences, Lund University
 defense location
 MH:C, Matematikhuset, SÃ¶lvegatan 18, Lund, Sweden
 defense date
 20030905 10:15
 ISSN
 14040034
 language
 English
 LU publication?
 yes
 id
 4deb440cc0e644908b1055a28034b067 (old id 466076)
 date added to LUP
 20070927 16:12:17
 date last changed
 20180529 11:14:12
@phdthesis{4deb440cc0e644908b1055a28034b067, abstract = {Paper A investigates how to simulate a differentiated mean in cases where interchanging differentiation and expectation is not allowed. Three approaches are available, finite differences (FD's), infinitesimal perturbation analysis (IPA) and the likelihood ratio score function (LRSF) method. We study FD's under discontinuities and show that the optimal decay rate of the mean square error is typically like n^{4/5}. The IPA method is generalized to allow for random variables with a finite number of jumps. Finally, we give a unified view of IPA and LRSF, which shows that, in the setting we consider, they are actually identical as long as the mathematics goes.<br/><br> <br/><br> Paper B provides error rates related to simulation of a Levy process when small jumps have been truncated. Three different truncations schemes are considered. Question: How to chose the truncation threshold? Answer: Weak error rates. We also consider a process X of infinite variation and the finest approximation in more detail and give a general Edgeworth expansion for a triangular array of Levy processes with third Levy moment 0.<br/><br> <br/><br> Paper C treats the inhomogeneous stochastic Cauchy problem for the wave equation. The initial values and driving force are assumed to be given stochastic distribution valued functions. We show existence and uniqueness and the solution $U(t,x)=U(t,x,omega)$ satisfies the equation in the strong sense with respect to time and space parameters $t$ and $x$.<br/><br> <br/><br> Paper D deals with a class of nonlinear heat equations driven by the spacetime derivative of a Brownian sheet. An Eulerlike approximation scheme is applied in order to solve the equation numerically. We show that the approximation scheme converges uniformly in probability.<br/><br> <br/><br> Finally, the note on a paperinprogres in chapter E deals with a digital communication system considered in a setting where the bit error probabilitiy p is so small that crude Monte Carlo simulation is not feasible for evaluating p and further related characteristics.}, author = {Signahl, Mikael}, issn = {14040034}, keyword = {Mathematics,Matematik,stochastic heat equation,stochastic wave equation,error rates,truncated Levy process,discontinuities,optimal decay rate,digital communication system}, language = {eng}, pages = {137}, publisher = {Centre for Mathematical Sciences, Lund University}, school = {Lund University}, title = {Topics in Simulation and Stochastic Analysis}, year = {2003}, }