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Fast random sampling and small noise analysis for stochastic control models

Sarvesh Ravichandran Iyer, Vivek Kumar

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.

Fast random sampling and small noise analysis for stochastic control models

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 , as well as by white noise of small intensity . We study the behavior of the system as and 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 and . 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.
Paper Structure (22 sections, 47 theorems, 279 equations)

This paper contains 22 sections, 47 theorems, 279 equations.

Table of Contents

  1. Introduction
  2. The Novelty and Methods
  3. Plan of the paper
  4. Notations
  5. Problem Formulation, Assumptions and Main Results
  6. Preliminaries
  7. Classical Results from Literature
  8. Key estimates
  9. Proof of LLN type Results
  10. Proof of the CLT Theorem \ref{['CLTresult']}
  11. The Decomposition of Random Sampling Part: Proof of Proposition \ref{['L1asymptotes']}
  12. The G1 Term
  13. The G2 Term
  14. The G3 Term
  15. The G5 Term
  16. ...and 7 more sections

Key Result

Theorem 2.1

Let $x(t)$ denotes the solution of contrlUsoln and let $X^{\varepsilon,n}_t$ be the solution to SDE controlwithnoise. Let $T\geq 0$ be fixed. Then, for arbitrary $\varepsilon>0, n\in \mathbb N, p\geq 1,$ there exists $C_{ABKTp} > 0,$ depending only on $A, B, K$,$T$ and $p$ such that where $\mathcal{N}_p = \int_0^T (s-\pi^n(s))^p ds$ satisfies $\mathbb{E}[\mathcal{N}_p]\le \frac{1}{n^p}C_{T\xi_1}$

Theorems & Definitions (90)

  • Theorem 2.1
  • Theorem 2.2: Central Limit Type Theorem
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 3.1: Wald's Identity
  • Theorem 3.2: The elementary renewal theorem
  • Theorem 3.3: Donsker's theorem
  • Theorem 3.4: Donsker's theorem for renewal processes
  • ...and 80 more