Abstract

This thesis examines the long--run behaviour of both differential and difference, deterministic and stochastic linear Volterra equations. Firstly we consider a stationary autoregressive conditional heteroskedastic (ARCH) process of order infinity. This type of process is used in time series analysis due to its non-constant conditional variance. In describing the extent of the dependence of the current values of the process upon past values we are led to the study of the autocovariance function. Necessary and sufficient conditions are established for the autocovariance function to lie in a particular class of slowly decaying (subexponential) sequences. We develop sharp conditions for solutions of linear Volterra summation equations to lie in a class of sequences which is characterised by having a subexponential rate of decay coupled with a periodic fluctuation. This theory illustrates and clarifies the effect of the kernel upon the solution of Volterra-type summation equations. In particular this theory is applied to the autocovariance function of ARCH infinity processes. A stochastic admissibility theory of stochastic Volterra operators is developed. In particular necessary and sufficient conditions for mean square convergence and sufficient conditions for almost sure convergence are established for stochastic integrals. This theory is then applied to stochastic linear functional equations of Volterra and finite delay type. Lastly, we introduce a particular stochastic differential equation with an average functional which may be viewed as modelling the demand of traders in an inefficient financial market. The asymptotic behaviour of this process is determined for almost all values of the parameters of the model. A discretisation of this stochastic differential equation is also studied. The asymptotic behaviour of the discretisation is shown to mirror that of the continuous-time equation.