The tail process and tail measure of continuous time regularly varying stochastic processes

Abstract

The goal of this paper is to investigate the tools of extreme value theory originally introduced for discrete time stationary stochastic processes (time series), namely the tail process and the tail measure, in the framework of continuous time stochastic processes with paths in the space $\mathcal{D}$ of c\`adl\`ag functions indexed by $\mathbb{R}$, endowed with Skorohod's $J_1$ topology. We prove that the essential properties of these objects are preserved, with some minor (though interesting) differences arising. We first obtain structural results which provide representation for homogeneous shift-invariant measures on $\mathcal{D}$ and then study regular variation of random elements in $\mathcal{D}$. We give practical conditions and study several examples, recovering and extending known results.