Extremal behaviour of solutions to a stochastic difference equation with applications to ARCH processes
type: Item
instance of: scholarly article
title: Extremal behaviour of solutions to a stochastic difference equation with applications to ARCH processes
zbMATH Open document ID: 0679.60029
DOI: 10.1016/0304-4149(89)90076-8
author: Holger Rootzén
author: Casper G. de Vries
author: Laurens De Haan
author: Sidney I. Resnick
published in: Stochastic Processes and their Applications
publication date: 1989-01-01T00:00:00Z
review text: Let \(\{Y_ n\}\), \(n\geq 1\), be a sequence of random variables such that \(Y_ n=A_ nY_{n-1}+B_ n\), \(n\geq 1\), \(Y_ 0\geq 0\), where \((A_ n,B_ n)'s\) are i.i.d. \({\mathbb{R}}^ 2_+\)-valued random pairs. An example of such a sequence is \(\{X^ 2_ n\}\) where \(\{X_ n\}\) is the first order autoregressive conditional heteroscedastic (ARCH) process used in econometric modelling. Now let \(M_ n=\max (Y_ 1,...,Y_ n)\) and for any \(C\subset {\mathbb{R}}_+\) define \(N_ n(C)=\#\{(k/n)\in C:\) \(Y_ k>u_ n\}\) where \(u_ n\) is a suitably chosen high level. In addition to some mild conditions on \((A_ 1,B_ 1)\), suppose there exists an \(\alpha >0\) such that \(E(A_ 1^{\alpha})=1\), and let \(u_ n=xn^{1/\alpha}\). Main results are: (1) \(P(M_ n\leq u_ n)\to \exp (-c\theta x^{-\alpha})\), \(x>0\), as \(n\to \infty\) and (2) \(N_ n\to^{d}N\), as \(n\to \infty\), where N is a compound Poisson process with intensity parameter \(c\theta x^{- \alpha}.\) Expressions for \(\theta\) and the compounding probabilities of N are given. Simulation procedures to compute them are described. They are applied to compute these parameters for the ARCH process. (The authors remark that the value of c is known only for integer values of \(\alpha\).)
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60H99
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 60G17
zbMATH DE Number: 4113646
zbMATH Keywords: extreme values
zbMATH Keywords: stochastic difference equation
zbMATH Keywords: ARCH process
zbMATH Keywords: econometric modelling
zbMATH Keywords: Simulation procedures
MaRDI profile type: Publication
full work available at URL: 0304-4149(89)90076-8
OpenAlex ID: W2035995445
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description: scientific article; zbMATH DE number 4113646