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

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

2008-11-13

arxiv.org/abs/0811.2052v1

Stochastic Processes and Applications 119, 10 (2009) 3583-3610

Mathematics

Probability

Scientific paper

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.

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