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Moderate Deviation Principles for Stochastic Differential Equations in Fast-Varying Markovian Environment

Hongjiang Qian

TL;DR

This work establishes a moderate deviation principle for a fully coupled two-time-scale system where a slow diffusive component X^ε evolves in a fast, finite-state jump environment Y^ε. Using a weak convergence framework combined with Poisson-equation methods for the fast process, the authors derive a Laplace principle with speed h^{-2}(ε) and identify a good rate function I(η) (with an equivalent Ĩ(η)) governing the deviations of X^ε from its averaged limit. The rate function is expressed via an optimal-control formulation involving Brownian and jump controls and employs a Poisson-equation-based representation to couple slow and fast dynamics. The paper also proves the compactness of level sets, analyzes limiting controlled dynamics, and establishes both lower and upper Laplace bounds to obtain the full MDP, including an extension to unbounded drift. The discussions outline extensions to jump-diffusion fast environments, countable-state fast environments with substantial technical challenges, and non-Markovian or more general fast dynamics, highlighting future directions and limitations.

Abstract

In this paper, we proved moderate deviation principles for a fully coupled two-time-scale stochastic systems, where the slow process is given by stochastic differential equations with small noise, while the fast process is a rapidly changing purely jump process on finite state space. The system is fully coupled in that the drift and diffusion coefficients of the slow process, as well as the jump distribution of the fast process, depend on states of both processes. Moreover, the diffusion component in the slow process can be degenerate. Our approach is based on the combination of the weak convergence method from [A. Budhiraja, P. Dupuis, and A. Ganguly, Electron. J. Probab. 23 (2018), pp. 1-33; Ann. Probab. 44 (2016), pp. 1723-1775] with Poisson equation for the fast-varying purely jump process.

Moderate Deviation Principles for Stochastic Differential Equations in Fast-Varying Markovian Environment

TL;DR

This work establishes a moderate deviation principle for a fully coupled two-time-scale system where a slow diffusive component X^ε evolves in a fast, finite-state jump environment Y^ε. Using a weak convergence framework combined with Poisson-equation methods for the fast process, the authors derive a Laplace principle with speed h^{-2}(ε) and identify a good rate function I(η) (with an equivalent Ĩ(η)) governing the deviations of X^ε from its averaged limit. The rate function is expressed via an optimal-control formulation involving Brownian and jump controls and employs a Poisson-equation-based representation to couple slow and fast dynamics. The paper also proves the compactness of level sets, analyzes limiting controlled dynamics, and establishes both lower and upper Laplace bounds to obtain the full MDP, including an extension to unbounded drift. The discussions outline extensions to jump-diffusion fast environments, countable-state fast environments with substantial technical challenges, and non-Markovian or more general fast dynamics, highlighting future directions and limitations.

Abstract

In this paper, we proved moderate deviation principles for a fully coupled two-time-scale stochastic systems, where the slow process is given by stochastic differential equations with small noise, while the fast process is a rapidly changing purely jump process on finite state space. The system is fully coupled in that the drift and diffusion coefficients of the slow process, as well as the jump distribution of the fast process, depend on states of both processes. Moreover, the diffusion component in the slow process can be degenerate. Our approach is based on the combination of the weak convergence method from [A. Budhiraja, P. Dupuis, and A. Ganguly, Electron. J. Probab. 23 (2018), pp. 1-33; Ann. Probab. 44 (2016), pp. 1723-1775] with Poisson equation for the fast-varying purely jump process.

Paper Structure

This paper contains 14 sections, 23 theorems, 247 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Main results
  4. The equivalent form of the rate function and its goodness
  5. An equivalent representation of the rate function
  6. Compact level sets
  7. Limiting behaviors of controlled processes
  8. Large deviation lower bound
  9. Large deviation upper bound
  10. Extension to unbounded drift
  11. Discussions and remarks
  12. Jump diffusions in fast-varying Markovian environment
  13. Countable state Markovian environment
  14. Beyond Markovian environment

Key Result

Theorem 2.2

Let $F$ be a bounded and measurable real-valued function, we have where $\vartheta\in (0,\infty)$ is a constant and $N^\vartheta :=(N_{ij}^\vartheta)_{(i,j)\in {\mathbb T}}$ with $N_{ij}^\vartheta$ defined under N-phi.

Theorems & Definitions (45)

  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Theorem 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 35 more