Asymptotic analysis of dynamical systems driven by Poisson random measures with periodic sampling

TL;DR

This paper analyzes the asymptotic behavior of nonlinear dynamical systems influenced by combined periodic sampling and small jump noise, revealing convergence to deterministic dynamics and characterizing fluctuations in different asymptotic regimes.

Contribution

It provides a detailed asymptotic analysis of stochastic systems with periodic sampling and jump noise, including convergence results and perturbation expansions.

Findings

Stochastic process converges to deterministic dynamics as noise and sampling intervals vanish.

Rescaled fluctuations converge to an effective process with an additional drift term.

Derived error bounds for the perturbation expansion of the stochastic process.

Abstract

In this article, we study the dynamics of a nonlinear system governed by an ordinary differential equation under the combined influence of fast periodic sampling with period $\delta$ and small jump noise of size $\varepsilon, 0< \varepsilon,\delta \ll 1.$ The noise is a combination of Brownian motion and Poisson random measure. The instantaneous rate of change of the state depends not only on its current value but on the most recent measurement of the state, as the state is measured at certain discrete-time instants. As $\varepsilon,\delta \searrow 0,$ the stochastic process of interest converges, in a suitable sense, to the dynamics of the deterministic equation. Next, the study of rescaled fluctuations of the stochastic process around its mean is found to vary depending on the relative rates of convergence of small parameters $\varepsilon, \delta$ in different asymptotic regimes. We show that the rescaled process converges, in a strong (path-wise) sense, to an effective process having an extra drift term capturing both the sampling and noise effect. Consequently, we obtain a first-order perturbation expansion of the stochastic process of interest, in terms of the effective process along with error bounds on the remainder.