On the Asymptotic Behaviour of Deterministic and Stochastic Volterra Integro--Differential Equations

Abstract

This thesis examines a question of stability in stochastic and deterministic systems with memory, and involves studying the asymptotic properties of Volterra integro-differential equations. The type of stability that has been established for this class of equations is important in a variety of real-world problems which involve feedback from the past, and are subject to external random forces. These include modelling endemic diseases, and more particularly the modelling of inefficient financial markets. The theine of the thesis is to subject a dynamical system with memory to increasingly trong and unpredictable external noise. Firstly, a fundamental deterministic Volterra quation is considered. Necessary and sufficient conditions for the solution to approach nontrivial limit are known. A strengthened version of these conditions is shown to be necessary and sufficient for exponential convergence to a nontrivial limit. Next, a Volterra equation with a fading stochastic perturbation is studied. Two types f stochastic convergence are considered: mean square and almost sure convergence. Conditions re found which ensure that the solution converges to a non-equilibrium random imit. Moreover, the rate at which this limit is approached is established. In the mean quare case, necessary and sufficient conditions on the resolvent, kernel and noise are determined o ensure this rate of convergence. In the almost sure case, the same conditions re found to be sufficient; furthermore, it is shown that the conditions on the resolvent and he kernel are necessary. A correspoilding result was also found to hold for a more general lass of weakly singular kernels. As in the deterministic case, necessary and sufficient onditions for the solution to converge exponentially fast to its limit are found. Finally, a stochastic Volterra equation with constant noise intensity is considered. This ives rise to the process analogous to Brownian motion, which has applications to mathematical inance. It can be shown that the increments of the process converge to a stationary tatistical distribution, which is Gaussian distributed. The conditions under which uch convergence can take place are completely characterised. In fact, a solution of a orresponding Volterra equation with infinite memory is shown to have exactly stationary ncrements which match the limiting distributions of the increments of solutions.