Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

A functional limit theorem for irregular SDEs

Stefan Ankirchner, Thomas Kruse, and Mikhail Urusov

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Abstract

Let $X_{1},X_{2},\ldots$ be a sequence of i.i.d. real-valued random variables with mean zero, and consider the scaled random walk of the form $Y^{N}_{k+1}=Y^{N}_{k}+a_{N}(Y^{N}_{k})X_{k+1}$, where $a_{N}:\mathbb{R}\to\mathbb{R}_{+}$. We show, under mild assumptions on the law of $X_{i}$, that one can choose the scale factor $a_{N}$ in such a way that the process $(Y^{N}_{\lfloor Nt\rfloor})_{t\in\mathbb{R}_{+}}$ converges in distribution to a given diffusion $(M_{t})_{t\in\mathbb{R}_{+}}$ solving a stochastic differential equation with possibly irregular coefficients, as $N\to\infty$. To this end we embed the scaled random walks into the diffusion $M$ with a sequence of stopping times with expected time step $1/N$.

Résumé

Soit $X_{1},X_{2},\ldots$ une suite de variables aléatoires indépendantes avec espérance $E(X_{i})=0$, et $Y^{N}_{k+1}=Y^{N}_{k}+a_{N}(Y^{N}_{k})X_{k+1}$ une marche aléatoire renormalisée avec une fonction $a_{N}:\mathbb{R}\to\mathbb{R}_{+}$. On montre, sous certaines conditions légères sur la loi de $X_{i}$, que l’on peut choisir le facteur $a_{N}$ d’une facon que $(Y^{N}_{\lfloor Nt\rfloor})_{t\in\mathbb{R}_{+}}$ converge en loi, quand $N$ tend vers l’infini, vers une diffusion $(M_{t})_{t\in\mathbb{R}_{+}}$ étant la solution d’une équation differentielle stochastique avec des coefficients irréguliers. À cet effet, nous plongeons la marche aléatoire renormalisée dans la diffusion $M$ par une suite de temps d’arrêt ayant un pas de temps avec espérance $1/N$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 3 (2017), 1438-1457.

Dates
Received: 15 July 2015
Revised: 27 February 2016
Accepted: 21 April 2016
First available in Project Euclid: 21 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1500624047

Digital Object Identifier
doi:10.1214/16-AIHP760

Mathematical Reviews number (MathSciNet)
MR3689973

Zentralblatt MATH identifier
1372.60043

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60J60: Diffusion processes [See also 58J65] 65C30: Stochastic differential and integral equations

Keywords
Stochastic differential equations Irregular diffusion coefficient Weak law of large numbers for u.i. arrays Weak convergence of processes Skorokhod embedding problem

Citation

Ankirchner, Stefan; Kruse, Thomas; Urusov, Mikhail. A functional limit theorem for irregular SDEs. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 3, 1438--1457. doi:10.1214/16-AIHP760. https://projecteuclid.org/euclid.aihp/1500624047


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