Large fluctuations of stochastic differential equations with regime switching: applications to simulation and finance

Abstract

This thesis deals with the asymptotic behaviour of various classes of stochastic differential equations (SDEs) and their discretisations. More specifically, it concerns the largest fluctuations of such equations by considering the rate of growth of the almost sure running maxima of the solutions. The first chapter gives a brief overview of the main ideas and motivations for this thesis. Chapter 2 examines a class of nonlinear finite–dimensional SDEs which have mean– reverting drift terms and bounded noise intensity or, by extension, unbounded noise intensity. Equations subject to Markovian switching are also studied, allowing the drift and diffusion coefficients to switch randomly according to a Markov jump process. The assumptions are motivated by the large fluctuations experienced by financial markets which are subjected to random regime shifts. We determine sharp upper and lower bounds on the rate of growth of the large fluctuations of the process by means of stochastic comparison methods and time change techniques. Chapter 3 applies similar techniques to a variant of the classical Geometric Brownian Motion (GBM) market model which is subject to random regime shifts. We prove that the model exhibits the same long–run growth properties and deviations from the trend rate of growth as conventional GBM. The fourth chapter examines the consistency of the asymptotic behaviour of a discretisation of the model detailed in Chapter 3. More specifically, it is shown that the discrete approximation to the stock price grows exponentially and that the large fluctuations from this exponential growth trend are governed by a Law of the Iterated Logarithm. The results about the asymptotic behaviour of discretised SDEs found in Chapter 4, rely on the use of an exponential martingale inequality (EMI). Chapter 5 considers a discrete version of the EMI driven by independent Gaussian sequences. Some extensions, applications and ramifications of the results are detailed. The final chapter uses the EMI developed in Chapter 5 to analyse the asymptotic behaviour of discretised SDEs. Two different methods of discretisation are considered: a standard Euler–Maruyama method and an implicit split–step variant of Euler–Maruyama. v