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Limit theorems for SDEs with irregular drifts

Jianhai Bao, Jiaqing Hao

Abstract

In this paper, concerning SDEs with Hölder continuous drifts, which are merely dissipative at infinity, and SDEs with piecewise continuous drifts, we investigate the strong law of large numbers and the central limit theorem for underlying additive functionals and reveal the corresponding rates of convergence. To establish the limit theorems under consideration, the exponentially contractive property of solution processes under the (quasi-)Wasserstein distance plays an indispensable role. In order to achieve such contractive property, which is new and interesting in its own right for SDEs with Hölder continuous drifts or piecewise continuous drifts, the reflection coupling method is employed and meanwhile a sophisticated test function is built.

Limit theorems for SDEs with irregular drifts

Abstract

In this paper, concerning SDEs with Hölder continuous drifts, which are merely dissipative at infinity, and SDEs with piecewise continuous drifts, we investigate the strong law of large numbers and the central limit theorem for underlying additive functionals and reveal the corresponding rates of convergence. To establish the limit theorems under consideration, the exponentially contractive property of solution processes under the (quasi-)Wasserstein distance plays an indispensable role. In order to achieve such contractive property, which is new and interesting in its own right for SDEs with Hölder continuous drifts or piecewise continuous drifts, the reflection coupling method is employed and meanwhile a sophisticated test function is built.
Paper Structure (7 sections, 6 theorems, 140 equations)

This paper contains 7 sections, 6 theorems, 140 equations.

Table of Contents

  1. Introduction and main results
  2. LLN for SDEs with Hölder continuous drifts: partial dissipativity
  3. CLT for SDEs with Hölder continuous drifts: monotone and Lyapunov conditions
  4. LLN and CLT for SDEs with piecewise continuous drifts
  5. Proof of Theorem \ref{['thm1']}
  6. Proof of Theorem \ref{['CLT']}
  7. Proof of Theorem \ref{['thm2']}

Key Result

Theorem 1.2

Assume $($${\bf H}_b$$)$ and $($${\bf H}_\sigma$$)$. Then, for any $f\in C_{\rm Lip}(\mathbb R^d)$ and $\varepsilon\in(0,1/2)$, there exist a random time $T_\varepsilon\ge 1$ and a constant $C>0$$($dependent on the Lipschitz constant $\|f\|_{\rm Lip}$ and the initial value $x$$)$ such that for all $ where $\mu\in\mathscr P(\mathbb R^d)$ stands for the unique invariant probability measure of $(X_t^

Theorems & Definitions (17)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Proposition 2.1
  • proof
  • ...and 7 more