A connection between extreme value theory and long time approximation of SDE's

TL;DR

This paper links extreme value theory with the long-term approximation of SDEs, showing how maxima of i.i.d. sequences relate to Euler schemes of jump SDEs and establishing a functional limit theorem.

Contribution

It introduces a novel connection between extreme value theory and the long-time behavior of SDEs via Euler schemes with decreasing steps.

Findings

Maxima of i.i.d. sequences can be modeled as Euler schemes of jump SDEs.

A functional limit theorem for maxima sequences is established.

The approach bridges extreme value theory and ergodic SDE approximation methods.

Abstract

We consider a sequence $(\xi_n)_{n\ge1}$ of $i.i.d.$ random values living in the domain of attraction of an extreme value distribution. For such sequence, there exists $(a_n)$ and $(b_n)$, with $a_n>0$ and $b_n\in\ER$ for every $n\ge 1$, such that the sequence $(X_n)$ defined by $X_n=(\max(\xi_1,...,\xi_n)-b_n)/a_n$ converges in distribution to a non degenerated distribution. In this paper, we show that $(X_n)$ can be viewed as an Euler scheme with decreasing step of an ergodic Markov process solution to a SDE with jumps and we derive a functional limit theorem for the sequence $(X_n)$ from some methods used in the long time numerical approximation of ergodic SDE's.