Topics in the numerical simulation of pathwise solutions to stochastic differential equations

• Main Author: Gaines, Jessica Gabrielle
• Published: University of Edinburgh 1995
• Subjects: 519
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.651182

This work contains several developments in the area of numerical solution of pathwise solutions to stochastic differential equations (SDE's). In the first chapter we define and motivate pathwise solutions and give a brief survey of numerical methods for approximating them. The main key to enlarging the scope of numerical methods for SDE's is a good representation of Brownian paths. A binary tree structure is an essential tool in Chapter Two, which presents a general method for solution of SDE's using variable time steps. In the case of a general SDE, improvement of the order of convergence compared with standard methods, demands generation of the Lévy area integrals. Chapter Three presents a method of random generation of the Lévy area for a Brownian path in <I>IR</I><SUP>2</SUP>. The method is based on Marsaglia's rectangle-wedge-tail method for fast generation of normally distributed deviates. Since the solution of an SDE generally depends on an infinite sequence of iterated integrals of the driving noise, it makes sense to examine these integrals and the algebraic relations between them. In Chapter Four, it is shown how known facts about shuffle algebras can be used to get a better understanding of stochastic iterated integrals of Ito and Stratonovich type and obtain practical algebraic bases for these two sets. We use the algebra to calculate moments of stochastic integrals, needed when calculating moments of error during numerical solutions of SDE's. The work on the generation of area integrals, described in Chapter Three, gives rise to general questions about the generation of random deviates, some of which are addressed in the last two chapters. In Chapter Five, we present a polynomial-time algorithm for finding the partition, into rectangles or triangles, of certain types of region in <I>IR</I><SUP>2</SUP>, that has the lowest entropy.