Hilfer fractional stochastic pantograph equations: Poisson jumps and the averaging principle with numerical simulation

Abstract This paper develops and analyzes an averaging technique for Hilfer fractional stochastic differential pantograph equations (HFSDPEs) driven by Poisson jumps. The study is conducted under non‐Lipschitz conditions, which significantly broadens the class of admissible systems compared to conventional assumptions. By employing stochastic analysis tools and fractional calculus, we demonstrate that the solutions of HFSDPEs can be approximated by the probability distribution of the corresponding averaged stochastic systems in the mean‐square sense. The novelty of this work lies in the integration of the Hilfer fractional derivative with stochastic pantograph structures influenced by discontinuous Poisson noise, a combination not previously treated within the averaging framework. This approach not only unifies fractional‐order memory effects with random jump perturbations but also extends existing averaging principles from classical and Caputo‐type systems to the more general Hilfer fractional setting. Furthermore, a numerical simulation of the pantograph system is presented to validate the theoretical results and illustrate the accuracy and efficiency of the proposed averaging scheme using the Gauss–Legendre quadrature rule. The findings thus provide a comprehensive theoretical–numerical framework for studying the stability and approximation of fractional stochastic systems with delay and impulsive effects.