[Paper Review] Moderate Deviation Principle for dynamical systems with small random perturbation

Yutao Ma, Ran Wang+1 more|arXiv (Cornell University)|Jul 18, 2011

Stochastic processes and financial applications9 references3 citations

TL;DR

This paper establishes a moderate deviation principle (MDP) for solutions of small noise stochastic differential equations (SDEs) in $\mathbb{R}^d$, extending Freidlin-Wentzell's large deviation theory. By analyzing the asymptotic behavior of $X^\varepsilon_t$ under a scaling where the noise intensity $\sqrt{\varepsilon}$ is neither vanishingly small nor large, the authors derive a variational formula for the MDP rate function, providing a refined approximation between large deviations and Gaussian fluctuations.

ABSTRACT

Consider the stochastic differential equation in $ r^d$ dX^{\e}_t&=b(X^{\e}_t)dt+\sqrt{\e}σ(X^\e_t)dB_t X^{\e}_0&=x_0,\quad x_0\in r^d$ where $b: r^d o r^d$ is $C^1$ such that $ \leq C(1+|x|^2)$, $σ: r^d o \MM(d imes n)$ is locally Lipschitzian with linear growth, and $B_t$ is a standard Brownian motion taking values in $ r^n$. Freidlin-Wentzell's theorem gives the large deviation principle for $X^\e$ for small $\e$. In this paper we establish its moderate deviation principle.

Motivation & Objective

• To extend Freidlin-Wentzell's large deviation principle to moderate deviations in stochastic dynamical systems with small random perturbations.
• To analyze the asymptotic behavior of solutions $X^\varepsilon_t$ when the noise intensity $\sqrt{\varepsilon}$ scales moderately, not vanishingly.
• To derive a variational formula for the moderate deviation rate function using the action functional of the underlying SDE.

Proposed method

• Formulate the stochastic differential equation $dX^\varepsilon_t = b(X^\varepsilon_t)dt + \sqrt{\varepsilon}\sigma(X^\varepsilon_t)dB_t$ with $C^1$ drift and locally Lipschitz diffusion coefficients.
• Apply the weak convergence approach to prove the moderate deviation principle by analyzing the convergence of controlled processes.
• Use the Donsker-Varadhan variational formula to characterize the rate function in terms of the action functional of the controlled SDE.
• Establish tightness and exponential equivalence of the controlled processes to handle the nonlinearities in drift and diffusion coefficients.
• Verify the necessary conditions on the drift $b$ and diffusion $\sigma$ to ensure the existence and regularity of the solution and its controlled version.
• Derive the moderate deviation rate function as the solution to a variational problem involving the energy of the control process.

Experimental results

Research questions

• RQ1 How does the moderate deviation behavior of an SDE with small noise differ from both the large deviation and central limit theorem regimes?
• RQ2 What is the precise form of the rate function governing moderate deviations in this class of SDEs?
• RQ3 Can the weak convergence method be adapted to derive the moderate deviation principle under the given regularity assumptions on $b$ and $\sigma$?
• RQ4 How does the structure of the action functional relate to the optimal control path in the moderate deviation scaling?
• RQ5 What conditions on $b$ and $\sigma$ ensure the validity of the moderate deviation principle in the $C^1$ and linear growth setting?

Key findings

• The moderate deviation principle holds for the SDE under the stated $C^1$ drift and locally Lipschitz diffusion with linear growth conditions.
• The rate function for the moderate deviation principle is characterized as the infimum of the action functional over all absolutely continuous paths with finite energy.
• The rate function is expressed via a variational formula involving the controlled SDE, generalizing the Freidlin-Wentzell framework.
• The convergence of the controlled processes is established via tightness and exponential equivalence, ensuring the validity of the MDP.
• The result provides a bridge between large deviations and Gaussian fluctuations, offering a refined approximation for moderate-scale deviations.