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Topics in Simulation and Stochastic Analysis

Signahl, Mikael LU (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 space-time derivative of a Brownian sheet. An Euler-like 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 paper-in-progres 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:
author
opponent
  • Professor Kaj, Ingemar, Uppsala
organization
publishing date
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, Sölvegatan 18, Lund, Sweden
defense date
2003-09-05 10:15
ISSN
1404-0034
language
English
LU publication?
yes
id
4deb440c-c0e6-4490-8b10-55a28034b067 (old id 466076)
date added to LUP
2007-09-27 16:12:17
date last changed
2016-09-19 08:44:59
@phdthesis{4deb440c-c0e6-4490-8b10-55a28034b067,
  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 space-time derivative of a Brownian sheet. An Euler-like 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 paper-in-progres 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         = {1404-0034},
  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},
}