Second Moment Boundedness of Linear Stochastic Delay Differential Equations

TL;DR

This paper develops a framework to analyze the second moment boundedness of solutions to linear stochastic delay differential equations, providing explicit characteristic functions and conditions for boundedness.

Contribution

It introduces a general method to compute the characteristic function for second moment boundedness in N-dimensional linear stochastic delay equations and applies it to a specific 2D case.

Findings

Explicit characteristic function for second moment boundedness.

Sufficient conditions for boundedness or unboundedness.

Application to a 2D stochastic delay differential equation.

Abstract

This paper studies the second moment boundedness of solutions of linear stochastic delay differential equations. First, we give a framework, for general $\mathrm{N}$-dimensional linear stochastic differential equations with a single discrete delay, of calculating the characteristic function for the second moment boundedness. Next, we apply the proposed framework to a special case of a type of 2-dimensional equation that the stochastic terms are decoupled. For the 2-dimensional equation, we obtain the characteristic function explicitly given by equation coefficients, the characteristic function gives sufficient conditions for the second moment to be bounded or unbounded.