Exponential growth rate for a singular linear stochastic delay differential equation

TL;DR

This paper proves the existence of a deterministic exponential growth rate for solutions of a singular linear stochastic delay differential equation, independent of initial conditions, using invariant measures and a Furstenberg-Hasminskii-type formula.

Contribution

It introduces a novel approach to analyze the growth rate of singular stochastic delay equations, which was not covered by existing results.

Findings

Existence of a deterministic exponential growth rate for the solution norm.

Independence of the growth rate from initial conditions.

Development of a Furstenberg-Hasminskii-type formula for the 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 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).