Uniform-in-time bounds for a stochastic hybrid system with fast periodic sampling and small white-noise

TL;DR

This paper analyzes the long-term behavior of a nonlinear stochastic hybrid system under rapid periodic sampling and small noise, deriving uniform-in-time LLN and CLT results with explicit fluctuation control.

Contribution

It provides the first uniform-in-time LLN and CLT results for a non-Markovian hybrid system with fast sampling and small noise, including explicit fluctuation expansions.

Findings

The stochastic process closely follows an ODE as noise and sampling intervals vanish.

Fluctuations are controlled uniformly in time and include a drift term from sampling and noise effects.

Simulation results support the theoretical convergence and fluctuation estimates.

Abstract

We study the asymptotic behavior, uniform-in-time, of a non-linear dynamical system under the combined effects of fast periodic sampling with period $\delta$ and small white noise of size $\varepsilon,\thinspace 0<\varepsilon,\delta \ll 1$. The dynamics depend on both the current and recent measurements of the state, and as such it is not Markovian. Our main results can be interpreted as Law of Large Numbers (LLN) and Central Limit Theorem (CLT) type results. LLN type result shows that the resulting stochastic process is close to an ordinary differential equation (ODE) uniformly in time as $\varepsilon,\delta \searrow 0.$ Further, in regards to CLT, we provide quantitative and uniform-in-time control of the fluctuations process. The interaction of the small parameters provides an additional drift term in the limiting fluctuations, which captures both the sampling and noise effects. As a consequence, we obtain a first-order perturbation expansion of the stochastic process along with time-independent estimates on the remainder. The zeroth- and first-order terms in the expansion are given by an ODE and SDE, respectively. Simulation studies that illustrate and supplement the theoretical results are also provided.