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

Fabien Panloup (1, 2)

1 LSP - Laboratoire de Statistique et Probabilités
2 IMT - Institut de Mathématiques de Toulouse UMR5219

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 $X_n=(\max(\xi_1,\ldots,\xi_n)-b_n)/a_n$ 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.

Mots clés: stochastic differential equation; jump process; invariant distribution; Euler scheme; extreme value

Domaines: Probabilités [math.PR]

Fabien Panloup. A connection between extreme value theory and long time approximation of SDE's. Stochastic Processes and their Applications, 2009, 119 (10), pp.3583-3610. ⟨hal-00338401⟩