On the Almost Sure Partial Maxima of Solutions of Affine Stochastic Functional Differential Equations with Finite Delay

This paper studies the large fluctuations of solutions of scalar and finite–dimensional affine stochastic functional differential equations with finite memory, as well as related nonlinear equations. We find conditions under which the exact almost sure growth rate of the partial maximum of each component of the system can be determined, both for affine and nonlinear equations. The proofs exploit the fact that an exponentially decaying fundamental solution of the underlying deterministic equation is sufficient to ensure that the solution of the affine equation converges to a stationary Gaussian process.