Exponential growth rate for a singular linear stochastic delay differential equation

Abstract

We establish the existence of a deterministic exponential growth rate for the norm (on an appropriate function space)of the solution of the linear scalar stochastic delay equation $d X(t) = X(t-1) d W(t)$ which does not depend on the initial condition as longas it is not identically zero. Due to the singular nature of the equation this property does not follow fromavailable results on stochastic delay differential equations. The key technique is to establish existence and uniquenessof an invariant measure of the projection of the solution onto the unit sphere in the chosen function space via asymptotic coupling and to prove a Furstenberg-Hasminskii-type formula (like in the finite dimensional case).