Convergence of diffusions and their discretizations: from continuous to discrete processes and back

TL;DR

This paper develops new quantitative bounds for the convergence of diffusion processes and their discretizations, bridging continuous and discrete stochastic models with implications for geometric convergence analysis.

Contribution

It introduces explicit convergence bounds under mild conditions and offers a novel approach to analyze diffusion convergence via discretization sequences.

Findings

Established new convergence bounds for autoregressive models

Proved geometric convergence of Euler-Maruyama discretizations

Provided a novel method for analyzing diffusion processes through discretizations

Abstract

In this paper, we establish new quantitative convergence bounds for a class of functional autoregressive models in weighted total variation metrics. To derive our results, we show that under mild assumptions, explicit minorization and Foster-Lyapunov drift conditions hold. The main applications and consequences of the bounds we obtain concern the geometric convergence of Euler-Maruyama discretizations of diffusions with identity covariance matrix. Second, as a corollary, we provide a new approach to establish quantitative convergence of these diffusion processes by applying our conclusions in the discrete-time setting to a well-suited sequence of discretizations whose associated stepsizes decrease towards zero.