[Paper Review] Deterministic homogenization for discrete-time fast-slow systems under optimal moment assumptions

Ilya Chevyrev, Peter K. Friz+3 more|arXiv (Cornell University)|Mar 25, 2019

Advanced Mathematical Modeling in Engineering4 citations

TL;DR

This paper establishes deterministic homogenization for discrete-time fast-slow systems under optimal moment conditions using p-variation rough path theory. It proves weak convergence of the slow component to an Itô diffusion with explicitly characterized drift and diffusion coefficients, extending prior continuous-time results to discrete time with minimal moment assumptions.

ABSTRACT

We consider discrete-time fast-slow systems of the form $$ X^{(n)}_{k+1} = X^{(n)}_k + n^{-1}a_n(X_k^{(n)},Y_k^{(n)}) + n^{-1/2}b_n(X_k^{(n)},Y_k^{(n)})\;, \quad Y_{k+1}^{(n)} = T_nY_k^{(n)}\;.$$ We give conditions under which the dynamics of the slow equations converge weakly to an Ito diffusion $X$ as $n o\infty$. The drift and diffusion coefficients of the limiting stochastic differential equation satisfied by $X$ are given explicitly. This extends the results of [Kelly--Melbourne, J. Funct. Anal. 272 (2017) 4063-4102] from the continuous-time case to the discrete-time case. Moreover, our methods ($p$-variation rough paths) work under optimal moment assumptions.

Motivation & Objective

• To extend deterministic homogenization results from continuous-time to discrete-time fast-slow systems.
• To establish weak convergence of the slow component to an Itô diffusion under minimal moment assumptions.
• To derive explicit expressions for the drift and diffusion coefficients of the limiting SDE.
• To demonstrate the effectiveness of p-variation rough path techniques in discrete-time settings with optimal moment conditions.

Proposed method

• Analyzes discrete-time fast-slow systems of the form $ X^{(n)}_{k+1} = X^{(n)}_k + n^{-1}a_n(X_k^{(n)},Y_k^{(n)}) + n^{-1/2}b_n(X_k^{(n)},Y_k^{(n)}) $, $ Y_{k+1}^{(n)} = T_nY_k^{(n)} $.
• Applies p-variation rough path theory to handle the discrete-time dynamics and control the error in homogenization.
• Imposes moment conditions on the coefficients $ a_n $ and $ b_n $ that are optimal for convergence.
• Uses the ergodicity and mixing properties of the fast process $ Y_k^{(n)} $ to derive effective coefficients for the limiting diffusion.
• Derives the limiting Itô SDE by computing the averaged drift and diffusion coefficients via statistical properties of the fast process.
• Establishes weak convergence of the slow process $ X^{(n)} $ to the solution of the limiting Itô SDE as $ n \to \infty $.

Experimental results

Research questions

• RQ1 Under what conditions does the slow component of a discrete-time fast-slow system converge weakly to a diffusion process?
• RQ2 How can the drift and diffusion coefficients of the limiting Itô SDE be explicitly characterized in the discrete-time setting?
• RQ3 What moment assumptions on the coefficients $ a_n $ and $ b_n $ are necessary and sufficient for homogenization in discrete time?
• RQ4 Can p-variation rough path techniques be adapted to achieve optimal moment conditions in discrete-time homogenization?
• RQ5 How does the discrete-time homogenization result compare to existing continuous-time results in terms of generality and assumptions?

Key findings

• The slow component $ X^{(n)} $ converges weakly to an Itô diffusion as $ n \to \infty $, under optimal moment conditions on the coefficients.
• The drift and diffusion coefficients of the limiting SDE are explicitly computed as averages over the invariant measure of the fast process $ Y_k^{(n)} $.
• The convergence is established using p-variation rough path theory, which allows for minimal moment assumptions.
• The framework extends the results of Kelly and Melbourne (2017) from continuous-time to discrete-time systems.
• The method achieves homogenization without requiring higher-than-second-order moments, confirming optimality of the assumptions.
• The limiting SDE is well-defined and the homogenization limit is deterministic in the sense that the effective coefficients are non-random and explicitly computable.