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

Fabien Panloup

paper

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

arXiv:0811.2052

Abstract

We consider a sequence $(ξ_{n})_{n \geq 1}$ 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 $n \geq 1$ , such that the sequence $(X_{n})$ defined by $X_{n} = (max (ξ_{1}, ..., ξ_{n}) - b_{n}) / a_{n}$ 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.