[Paper Review] Non-Asymptotic Gaussian Estimates for the Recursive Approximation of the Invariant Measure of a Diffusion

Igor Honoré, Stéphane Menozzi+1 more|arXiv (Cornell University)|May 27, 2016

Stochastic processes and financial applications25 references3 citations

TL;DR

This paper establishes non-asymptotic Gaussian concentration bounds for the deviation between the invariant distribution of an ergodic diffusion and the empirical measure of a recursive Euler-type scheme with decreasing time steps. It derives computable confidence intervals and applies the results to non-asymptotic deviation bounds for the almost sure Central Limit Theorem, under conditions involving coboundary structure and Lyapunov functions.

ABSTRACT

We obtain non-asymptotic Gaussian concentration bounds for the difference between the invariant measure $ν$ of an ergodic Brownian diffusion process and the empirical distribution of an approximating scheme with decreasing time step along a suitable class of (smooth enough) test functions f such that f -- $ν$(f) is a coboundary of the infinitesimal generator. We show that these bounds can still be improved when the (squared) Fr{ö}benius norm of the diffusion coefficient lies in this class. We apply these bounds to design computable non-asymptotic confidence intervals for the approximating scheme. As a theoretical application, we finally derive non-asymptotic deviation bounds for the almost sure Central Limit Theorem.

Motivation & Objective

• To derive non-asymptotic concentration bounds for the difference between the invariant distribution ν and the empirical measure νₙ of a recursive scheme.
• To design computable, non-asymptotic confidence intervals for the scheme’s empirical mean under a coboundary structure for test functions.
• To extend the almost sure Central Limit Theorem with non-asymptotic deviation bounds using the proposed concentration estimates.
• To improve bounds when squared norms of the diffusion coefficient lie in the same function class as the test functions.
• To establish theoretical guarantees for the convergence rate of the empirical measure to the invariant distribution without bias from constant-step schemes.

Proposed method

• The authors use a recursive Euler-type discretization with decreasing time steps (γₖ) and innovations matching Gaussian moments up to order three.
• They analyze the deviation νₙ(f) − ν(f) for test functions f such that f − ν(f) is a coboundary of the infinitesimal generator A.
• The key technique involves solving the Poisson equation Aφ = f − ν(f) and using the resulting φ to control the deviation via martingale and Lyapunov function arguments.
• They derive Gaussian concentration bounds using a Lyapunov function V satisfying AV ≤ β − αV with α > 0, ensuring ergodicity and moment bounds.
• The bounds are refined when |σ∇φ|² lies in the same class as f − ν(f), allowing tighter variance estimates.
• Numerical validation is performed via Monte Carlo simulations to assess the tightness of the bounds and confidence interval performance.

Experimental results

Research questions

• RQ1 Can non-asymptotic Gaussian concentration bounds be established for the empirical measure of a decreasing-step Euler scheme approximating the invariant distribution of a diffusion?
• RQ2 How can these bounds be improved when the squared norm of the diffusion coefficient’s gradient lies in the same function class as the test function?
• RQ3 What is the rate of convergence of the empirical measure to the invariant distribution in terms of deviation probabilities?
• RQ4 Can the derived bounds be used to construct computable, non-asymptotic confidence intervals for the scheme’s empirical mean?
• RQ5 What non-asymptotic deviation bounds can be derived for the almost sure Central Limit Theorem using this framework?

Key findings

• Non-asymptotic Gaussian concentration bounds are established for νₙ(f) − ν(f) under the coboundary condition f − ν(f) = Aφ, with explicit dependence on the Lyapunov function and generator properties.
• The bounds are improved when |σ∇φ|² belongs to the same class as f − ν(f), leading to tighter variance estimates.
• Computable non-asymptotic confidence intervals are derived for the empirical mean, with a 95% confidence interval size of 4.75054×10⁻⁵ in simulations at n = 5×10⁴.
• Theoretical deviation bounds for the almost sure Central Limit Theorem are obtained, extending classical results to non-asymptotic regimes.
• Simulations confirm that the theoretical bound Sσ(a) = −a²α²/(2[f]₁²) remains above the empirical log-probability curve gₙσ(a), validating the concentration inequality.
• The method avoids bias from constant-step schemes and ensures almost sure convergence of νₙ to ν without requiring explicit knowledge of ν(f), estimated via ergodic averaging.