Extremes

, Volume 12, Issue 3, pp 265–296

Extremes of Lévy driven mixed MA processes with convolution equivalent distributions

Article

DOI: 10.1007/s10687-008-0079-x

Cite this article as:
Fasen, V. Extremes (2009) 12: 265. doi:10.1007/s10687-008-0079-x

Abstract

We investigate the extremal behavior of stationary mixed MA processes \(Y(t)=\int_{\mathbb{R}_+\times\mathbb{R}}f(r,t-s)\,d\,\Lambda(r,s)\) for t ≥ 0, where f is a deterministic function and Λ is an infinitely divisible and independently scattered random measure. Particular examples of mixed MA processes are superpositions of Ornstein-Uhlenbeck processes, applied as stochastic volatility models in Barndorff-Nielsen and Shephard (2001a). We assume that the finite dimensional distributions of Λ are in the class of convolution equivalent tails and in the maximum domain of attraction of the Gumbel distribution. On the one hand, we compute the tail behavior of Y(0) and supt ∈ [0,1]Y(t). On the other hand, we study the extremes of Y by means of marked point processes based on maxima of Y in random intervals. A complementary result guarantees the convergence of the running maxima of Y to the Gumbel distribution.

Keywords

Convolution equivalent distribution Extreme value theory MA process Marked point process Mixed MA process Point process Random measure Shot noise process Subexponential distribution SupOU process 

AMS 2000 Subject Classifications

Primary—60G70; Secondary—60F05, 60G10, 60G55 

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Center for Mathematical SciencesTechnische Universität MünchenGarchingGermany