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

TL;DR

The paper analyzes a nonlinear controlled SDE with rapid periodic sampling and small additive noise, establishing uniform-in-time Law of Large Numbers and Central Limit Theorem results. The LLN shows that the stochastic process $X_t^{\varepsilon,\delta}$ stays close to the deterministic trajectory $x_t$ governed by $\dot x_t=f(x_t)+\kappa(x_t)$, with time-uniform error bounds. The CLT characterizes fluctuations through $Z_t^{\varepsilon,\delta}=(X_t^{\varepsilon,\delta}-x_t)/\varepsilon$, which converge uniformly in time to a process $Z_t$ solving a linear SDE with an extra drift term that encodes the combined effects of sampling and noise; this drift depends on $\mathsf c=\lim_{\varepsilon\to0}(\delta/\varepsilon)$. A first-order perturbation expansion $X_t^{\varepsilon,\delta}=x_t+\varepsilon Z_t+o(\varepsilon)$ is thereby obtained with time-independent remainder bounds, and a Gaussian-approximation corollary is provided. The results are complemented by simulations and extended to a general multivariate setting with explicit conditions, illustrating robustness beyond the core assumptions.

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 $δ$ and small white noise of size $\varepsilon,\thinspace 0<\varepsilon,δ\ll 1$. 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 $\varepsilon,δ\searrow 0.$ 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.

Figures (4)

• Figure 1: For the non-linear model \ref{['E:example-nonlinear-2']}, varying effect of $\mathbb{E}[X_T^{\varepsilon,\delta}-x_T-\varepsilon Z_T]$ over $300$ sample paths with different values of $\varepsilon$ and two initial conditions $X_0^{\varepsilon,\delta}=-0.07,1.5$. Here, the error $\mathbb{E}[X_T^{\varepsilon,\delta}-x_T-\varepsilon Z_T]$ and time are represented on the vertical and horizontal axes, respectively.
• Figure 2: For the non-linear model \ref{['E:example-nonlinear-1']}, varying effect of $\mathbb{E}[X_T^{\varepsilon,\delta}-x_T-\varepsilon Z_T]$ over $300$ sample paths with different values of $\varepsilon$ and two initial conditions $X_0^{\varepsilon,\delta}=-0.07,1.5$.
• Figure 3: For the linear model \ref{['E:example-linear-1']}, varying effect of $\mathbb{E}[X_T^{\varepsilon,\delta}-x_T-\varepsilon Z_T]$ over $300$ sample paths with different values of $\varepsilon$. Here, the error $\mathbb{E}[X_T^{\varepsilon,\delta}-x_T-\varepsilon Z_T]$ and time are represented on the vertical and horizontal axes, respectively.
• Figure 4: For the non-linear model \ref{['E:example-linear-2']}, varying effect of $\mathbb{E}[X_T^{\varepsilon,\delta}-x_T-\varepsilon Z_T]$ over $300$ sample paths with different values of $\varepsilon$. Here, the error $\mathbb{E}[X_T^{\varepsilon,\delta}-x_T-\varepsilon Z_T]$ and time are represented on the vertical and horizontal axes, respectively.

Theorems & Definitions (54)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Theorem 2.5
• Theorem 2.6
• Remark 2.7
• Corollary 2.8
• proof
• Remark 2.9