Thesis defense “Speed of convergence of diffusion approximations”

Speaker : Eustache Besançon
Télécom-Paris
Date: 08/12/2020
Time: 2:00 pm - 5:00 pm
Location: LINCS Seminars room

Abstract

In many fields of interest, Markov processes are a primary modelisation tool for random processes. Unfortunately it is often necessary to use very large or even infinite dimension state spaces, making the exact analysis of the various characteristics of interest (stability, stationary law, hitting times of certain domains, etc.) of the process difficult or even impossible . For quite a time, thanks in particular to martingale theory, it has been possible to make use of approximations by brownian diffusions. This enables an approximate analysis of the initial problem.

The main drawback of this approach is that it does not measure the error made in this approximation. The purpose is to dévelop a theory of error calculation for diffusion approximations .

For some time, the developement of the Stein-Malliavin method has enabled to get some precision over speed of convergence in classical theorems such as the Donsker theorem (functionnal convergence  of a random walk towards the Brownian motion)  or in the generalisation of the Binomial Poisson approximation path by path.

In this work we intend to extend the development of this theory for Markovian processes such as those than can be found in queueing theory, in epidemiology or in other fields of application.

Starting from the representation of Markov processes as Poisson measures, we extend the method developped by Laurent Decreusefond and Laure Coutin to assess the speed of convergence in diffusion approximations . To do so, we extend the Stein-Malliavin method to vectors of  processes rather than a single process. The limit is a gaussian process changed in time. The Stein Malliavin method being mainly developped to calculate  convergence towards the standard Brownian motion, it is adapted to the problem of convergence towards a time changend process using linear approximation methods. We therefore make use of Gaussian analysis to assess the dependency between the various time periods and to functionnal analysis to elect the right probabilistic spaces.