Inhibiting effects of Ito-type Brownian noise on the blowup of solutions of nonlinear pantograph differential equation

Abstract

This article studies the influence of multiplicative noise on the blow-up behavior of solutions of a nonlinear deterministic pantograph differential equation. The deterministic equation is perturbed by an Ito-type white noise and the unbounded growth rate is examined along Osgood condition. It is established that if the noise scaling parameter in the state dependent diffusion term is sufficiently strong, the presence of noise ensures that the blow up of solutions of the resulting nonlinear stochastic
Pantograph delay differential equation is inhibited or prevented from occurring. However, the underlying deterministic differential equation, where noise is absent, will still admit solutions which blow up to infinity at finite time.