Admissibility of Stochastic Linear Volterra Operators

This work examines the long run behaviour of both differential and difference, deterministic and stochastic linear Volterra equations. It is shown how observable phenomena in inefficient financial markets may be present in the solutions of stochastic functional differential equations. Firstly, we consider a stationary autoregressive conditional heteroskedastic (ARCH) process of order infinity. Necessary and sufficient conditions are established for the autocovariance function to lie in a particular class of slowly decaying sequences. Secondly, an admissibility theory of stochastic Volterra operators is developed which characterises sufficient conditions for almost sure convergence of stochastic Volterra integrals. This theory is applied to find the exact asymptotic behaviour of the solutions of a class of affine stochastic Volterra equations. Lastly the exact asymptotic behaviour of a stochastic differential equation with an average functional is determined for all real values of the parameters of the equation. This equation may be viewed as modelling the demand of traders in an inefficient financial market. A discretisation of this stochastic differential equation is also studied.