Topics in Simulation and Stochastic Analysis
Mikael Signahl
Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology
2003
ISBN 9162857924
LUTFMS10212003

Abstract:

This thesis consists of four papers and a note treating problems related
to simulation and stochastic partial differential equations.


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 n4/5 .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 Lévy 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 Lévy processes with third Lévy 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,?) 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.


Keywords:

discontinuities, optimal decay rate, truncated Lévy process,
error rates, stochastic wave equation, stochastic heat equation, numerical
scheme, digital communication system




