Fast random sampling and small noise analysis for stochastic control models

TL;DR

This paper analyzes the convergence of stochastic control systems with rapid sampling and small noise, demonstrating how they approach deterministic models and deriving associated stochastic differential equations.

Contribution

It introduces a joint analysis of rapid sampling and small noise effects in linear and nonlinear control systems, extending existing theories.

Findings

Convergence of stochastic systems to deterministic analogues as sampling frequency increases and noise diminishes.

Derivation of stochastic differential equations describing fluctuations around deterministic trajectories.

Extension of analysis to nonlinear systems with multiplicative noise.

Abstract

In this paper, we study a linear control system with a given state feedback law. The system is influenced by rapid random sampling occurring at frequency $\frac 1n, n \in \mathbb N$, as well as by white noise of small intensity $\varepsilon \in (0, 1]$. We study the behavior of the system as $n \to \infty$ and $\varepsilon \searrow 0$ jointly, and prove that it converges to its ideal deterministic analogue. For the random fluctuations around its analogous deterministic trajectory, we obtain either stochastic differential equations or an ordinary differential equation depending on the joint behavior of $\varepsilon$ and $n$. Further, we extend this problem to a nonlinear system driven by multiplicative white noise, where the noise intensity is scaled by a small parameter. In this case, we again perform a similar analysis as in the linear case.