Long Memory and Financial Market Bubble Dynamics in Affine Stochastic Differential Equations with Average Functionals

In this paper we consider the growth, large fluctuations and memory properties of an affine stochastic functional differential equation with an average functional where the contributions of the average and instantaneous terms are parameterised. An asymptotic analysis of the solution of this equation is conducted for all values of the parameters of the equation. When solutions are recurrent, we show that the autocovariance function of the solution decays at a polynomial rate, even though the solution is asymptotically equal to another asymptotically stationary process whose autocovariance function decays exponentially. It is shown that when solutions grow, they do so at either a polynomial or exponential rate in time depending on the sign of a parameter of the model, modulo some exceptional parameter sets. On these exceptional sets, solutions are recurrent on the real line with large fluctuations consistent with the Law of the Iterated logarithm, or exhibit subexponential yet superpolynomial growth.