Topics in Simulation and Stochastic Analysis

Mikael Signahl

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology
2003

ISBN 91-628-5792-4
LUTFMS--1021--2003


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