Sufficient Conditions for Polynomial Asymptotic Behaviour of the Stochastic Pantograph Equation

TL;DR

This paper establishes conditions under which solutions of the stochastic pantograph equation exhibit polynomial or exponential asymptotic behavior, including growth and decay, in both mean and almost sure senses.

Contribution

It provides new sufficient conditions for polynomial and exponential asymptotic behaviors of solutions to stochastic pantograph equations with multiple delays.

Findings

Solutions can grow polynomially under certain parameter conditions.

Solutions decay polynomially when stronger conditions are met.

Exponential growth bounds are possible when polynomial bounds are not achievable.

Abstract

This paper studies the asymptotic growth and decay properties of solutions of the stochastic pantograph equation with multiplicative noise. We give sufficient conditions on the parameters for solutions to grow at a polynomial rate in $p$-th mean and in the almost sure sense. Under stronger conditions the solutions decay to zero with a polynomial rate in $p$-th mean and in the almost sure sense. When polynomial bounds cannot be achieved, we show for a different set of parameters that exponential growth bounds of solutions in $p$-th mean and an almost sure sense can be obtained. Analogous results are established for pantograph equations with several delays, and for general finite dimensional equations.