Abstract:Probabilistic properties of HARCH(k) processes, as special stochastic volatility models, are investigated. We present necessary and sufficient conditions for existence of a stationary version of a HARCH(k) process with finite (2£ )th moments,
Our approach is based on the general Markov chain techniques of (Meyn and Tweedie, 1990). The conditions are explicit in the case of second moments, and also in the case of 4th moments of the HARCH(2) process.We also deduce explicit necessary and explicit sufficient condi… Show more
“…[See also Goldie (1991), Vervaat (1979) and Embrechts, K üppelberg and Mikosch (1997).] This result is probably true in much greater generality [some evidence is in Embrechts, Samorodnitsky, Dacorogna and Muller (1996)] and provides a class of models where input variables are light tailed but output process marginals are heavy tailed. This is in contrast with the usual ARMA model assumptions where innovations are heavy tailed and hence process marginals are heavy tailed.…”
Section: Atandt Labs-research and Lawrence Berkeley National Laboratorymentioning
Huge data sets from the teletraffic industry exhibit many nonstandard characteristics such as heavy tails and long range dependence. Various estimation methods for heavy tailed time series with positive innovations are reviewed. These include parameter estimation and model identification methods for autoregressions and moving averages. Parameter estimation methods include those of Yule-Walker and the linear programming estimators of Feigin and Resnick as well estimators for tail heaviness such as the Hill estimator and the qq-estimator. Examples are given using call holding data and interarrivals between packet transmissions on a computer network. The limit theory makes heavy use of point process techniques and random set theory.
“…[See also Goldie (1991), Vervaat (1979) and Embrechts, K üppelberg and Mikosch (1997).] This result is probably true in much greater generality [some evidence is in Embrechts, Samorodnitsky, Dacorogna and Muller (1996)] and provides a class of models where input variables are light tailed but output process marginals are heavy tailed. This is in contrast with the usual ARMA model assumptions where innovations are heavy tailed and hence process marginals are heavy tailed.…”
Section: Atandt Labs-research and Lawrence Berkeley National Laboratorymentioning
Huge data sets from the teletraffic industry exhibit many nonstandard characteristics such as heavy tails and long range dependence. Various estimation methods for heavy tailed time series with positive innovations are reviewed. These include parameter estimation and model identification methods for autoregressions and moving averages. Parameter estimation methods include those of Yule-Walker and the linear programming estimators of Feigin and Resnick as well estimators for tail heaviness such as the Hill estimator and the qq-estimator. Examples are given using call holding data and interarrivals between packet transmissions on a computer network. The limit theory makes heavy use of point process techniques and random set theory.
“…We cannot, however, reject other explanations for this power law (e.g. Embrechts et al 1998;Podobnik et al 2000; see Appendix) because 25 data points is a small sample for investigating behaviour at the tails of a distribution. The lognormal distribution can be justified theoretically because population growth is a multiplicative process (Dennis & Patil 1988).…”
A number of investigators have invoked a cascading local interaction model to account for power‐law‐distributed fluctuations in ecological variables. Invoking such a model requires that species be tightly coupled, and that local interactions among species influence ecosystem dynamics over a broad range of scales. Here we reanalyse bird population data used by Keitt & Stanley (1998, Dynamics of North American breeding bird populations. Nature, 393, 257–260) to support a cascading local interaction model. We find that the power law they report can be attributed to mixing of lognormal distributions. More tentatively, we propose that mixing of distributions accounts for other empirical power laws reported in the ecological literature.
“…Participants were given high‐frequency data and discussed them openly. This inspired my work on heterogeneous auto‐regressive conditional heteroscedastic processes with Gena, Michel Dacorogna and Ulrich Müller (Embrechts et al , 1998); Rudolf Grübel and I later proved that these processes are heavy‐tailed (Embrechts & Grübel, 1999).…”
Section: Quantitative Risk Managementmentioning
confidence: 95%
“…Paul Embrechts (left), Thomas Mikosch (center) and Claudia Klüppelberg (right) working on their book at the Mathematisches Forschungsinstitut Oberwolfach, in Germany. Swan! 3 Over the years, I wrote several other papers with Gena, in particular when he visited me later on sabbatical (Embrechts & Samorodnitsky, 1995; Embrechts et al , 1998; Embrechts & Samorodnitsky, 2003; Embrechts et al , 2004).…”
Summary
Paul Embrechts was born in Schoten, Belgium, on 3 February 1953. He holds a Licentiaat in Mathematics from Universiteit Antwerpen (1975) and a DSc from Katholieke Universiteit Leuven (1979), where he was also a Research Assistant from 1975 to 1983. He then held a lectureship in Statistics at Imperial College, London (1983–1985) and was a Docent at Limburgs Universitair Centrum, Belgium (1985–1989) before joining ETH Zürich as a Full Professor of Mathematics in 1989, where he remained until his retirement as an Emeritus in 2018. A renowned specialist of extreme‐value theory and quantitative risk management, he authored or coauthored nearly 200 scientific papers and five books, including the highly influential ‘Modelling of Extremal Events for Insurance and Finance’ (Springer, 1997) and ‘Quantitative Risk Management: Concepts, Techniques and Tools’ (Princeton University Press, 2005, 2015). He served in numerous editorial capacities, notably as Editor‐in‐Chief of the ASTIN Bulletin (1996–2005). Praised for his natural leadership and exceptional communication skills, he helped to bridge the gap between academia and industry through the foundation of RiskLab Switzerland and his sustained leadership for nearly 20 years. He gave numerous prestigious invited and keynote lectures worldwide and served as a member of the board of, or consultant for, various banks, insurance companies and international regulatory authorities. His work was recognised through several visiting positions, including at the Oxford‐Man Institute, and many awards. He is, inter alia, an Elected Fellow of the Institute of Mathematical Statistics (1995) and the American Statistical Association (2014), an Honorary Fellow of the Institute and the Faculty of Actuaries (2000), Honorary Member of the Belgian (2010) and French (2015) Institute of Actuaries and was granted four honorary degrees (University of Waterloo, 2007; Heriot‐Watt University, 2011; Université catholique de Louvain, 2012; City, University of London, 2017). The following conversation took place in Paul's office at ETH Zürich, 17–18 December 2018.
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