2014/12/23 by Sebastian Engelke, Engelke, Sebastian, Zakhar Kabluchko +1
Economics, Econometrics and Finance · Mathematics · #60G10 #60G51 #60G55 #60G70 #FOS: Mathematics #Financial Risk and Volatility Modeling #Primary #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #secondary
paper · pdf · doi:10.48550/arxiv.1412.7444
openalex publication_date 2014/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study stationary max-stable processes \η(t)\colon t∈\mathbb R\ admitting a representation of the form η(t)=maxi∈\mathbb N(Ui+ Yi(t)), where ∑i=1∞ δUi is a Poisson point process on \mathbb R with intensity \rm e-u \rm d u, and Y1,Y2,… are i.i.d. copies of a process \Y(t)\colon t∈\mathbb R\ obtained by running a Lévy process for positive t and a dual Lévy process for negative t. We give a general construction of such Lévy-Brown-Resnick processes, where the restrictions of Y to the positive and negative half-axes are Lévy processes with random birth and killing times. We show that these max-stable processes appear as limits of suitably normalized pointwise maxima of the form Mn(t)=maxi=1,…,n ξi(sn+t), where ξ1,ξ2,… are i.i.d. Lévy processes and sn is a sequence such that sn∼ c log n with c>0. Also, we consider maxima of the form maxi=1,…,n Zi(t/log n), where Z1,Z2,… are i.i.d. Ornstein--Uhlenbeck processes driven by an α-stable noise with skewness parameter β=-1. After a linear normalization, we again obtain limiting max-stable processes of the above form. This gives a generalization of the results of Brown and Resnick [Extreme values of independent stochastic processes, J. Appl. Probab., 14 (1977), pp. 732--739] to the totally skewed α-stable case.