  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 dX(t) = X(t-1) dW(t) which does not depend on the
initial condition as long as it is not identically zero. Due to the singular
nature of the equation this property does not follow from available results on
stochastic delay differential equations. The key technique is to establish
existence and uniqueness of 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).

Probability