Moment Boundedness of Linear Stochastic Delay Differential Equation with Distributed Delay

TL;DR

This paper investigates the conditions under which solutions of linear stochastic delay differential equations with distributed delay have bounded moments, using characteristic functions derived via Laplace transforms.

Contribution

It provides new sufficient conditions for the second moment boundedness of such equations, linking stochastic terms with stability properties.

Findings

Derived the characteristic function using Laplace transform techniques.

Proposed sufficient conditions for second moment boundedness.

Linked stochastic delay differential equations' stability to their deterministic counterparts.

Abstract

This paper studies the moment boundedness of solutions of linear stochastic delay differential equations with distributed delay. For a linear stochastic delay differential equation, the first moment stability is known to be identical to that of the corresponding deterministic delay differential equation. However, boundedness of the second moment is complicated and depends on the stochastic terms. In this paper, the characteristic function of the equation is obtained through techniques of Laplace transform. From the characteristic equation, sufficient conditions for the second moment to be bounded or unbounded are proposed.