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In this article, we study the dynamics of a system governed by an ordinarydifferential equation under the combined influence of fast periodic samplingwith period $\delta$ and small jump noise of size $\varepsilon, 0<\varepsilon,\delta \ll 1.$ The noise is a combination of Brownian motion andPoisson random measure. The instantaneous rate of change of the state dependsnot only on its current value \textit{but} on the most recent measurement ofthe 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. The studyof rescaled fluctuations of the stochastic process around its mean is found tovary depending on the relative rates of convergence of small parameters$\varepsilon, \delta$ in different asymptotic regimes. We show that therescaled process converges, in a strong (path-wise) sense, to an effectiveprocess having an extra drift term capturing both the sampling and noiseeffect. Consequently, we obtain a first-order perturbation expansion of thestochastic process of interest, in terms of the effective process along witherror bounds on the remainder.

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