论文标题
由泊松随机测量驱动的动态系统的渐近分析，并定期采样
Asymptotic analysis of dynamical systems driven by Poisson random measures with periodic sampling
论文作者
论文摘要
在本文中，我们研究了一个非线性系统的动力学，该非线性系统受普通微分方程与快速周期性采样（$δ$）的综合影响和大小$ \ varepsilon的小跳跃噪声的综合影响，0 <\ varepsilon，Δ\ ll 1. $。国家的变化速率不仅取决于其当前价值，还取决于国家的最新测量，因为国家是在某些离散时间瞬间测量的。作为$ \ varepsilon，δ\ searrow 0，$从适当意义上的随机过程收敛于确定性方程的动力学。接下来，发现对随机过程围绕其平均值的重新波动的研究发现，根据不同参数的相对收敛速率$ \ varepsilon，δ$在不同的渐近方案中。我们表明，重新制定的过程从强烈的（途径）意义上收敛到一个有效的过程，该过程具有额外的漂移项，同时捕获了采样和噪声效应。因此，就有效过程以及其余部分的误差界限而言，我们获得了随机过程的一阶扰动扩展。
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 $δ$ and small jump noise of size $\varepsilon, 0< \varepsilon,δ\ll 1.$ 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 $\varepsilon,δ\searrow 0,$ 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 $\varepsilon, δ$ 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.
