Estimation of the characteristics of a Lévy process observed at arbitrary frequency

Donsker-type functional limit theorems are proved for empirical processes arising from discretely sampled increments of a univariate Lévy process. In the asymptotic regime the sampling frequencies increase to infinity and the limiting object is a Gaussian process that can be obtained from the composition of a Brownian motion with a covariance operator determined by the Lévy measure. The results are applied to derive the asymptotic distribution of natural estimators for the distribution function of the Lévy jump measure. As an application we deduce Kolmogorov-Smirnov type tests and confidence bands.MSC 2000 subject classification: Primary: 60F05; Secondary: 60G51, 62G05

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“…Moreover, from Theorem 1 by Kappus and Reiß (2010) we know that under our moment assumption on ν (for k = 0, 1, 2 and any δ > 0)…”

Section: Proof Of Theoremmentioning

confidence: 94%

High-frequency Donsker theorems for Lévy measures

Nickl

Reiß

Söhl

et al. 2015

Probab. Theory Relat. Fields

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“…By integrating these models into fossil sequences, we suspect that further fine-scale details of macroevolution can be elucidated. Moreover, in quantitative finance, where, for practical purposes, the "fossil record" of stock prices through time is perfectly kept, fine-scale dynamics of jump processes can be inferred (Barndorff-Nielsen et al 2012;Belomestny 2010;Kappus and Reiß 2010), suggesting that such power exists for suitably densely sampled fossil sequences. However, if rapid evolutionary changes are concentrated near speciation events, as in the classical theory of punctuated equilibrium (Gould and Eldredge 1977), signals for those adaptive shifts may be difficult to locate without a phylogenetic context among fossil sequences (Bokma 2008).…”

Section: Discussionmentioning

confidence: 99%

Punctuated evolution shaped modern vertebrate diversity

Landis

Schraiber

2017

Preprint

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The relative importance of different modes of evolution in shaping phenotypic diversity remains a hotly debated question. Fossil data suggest that stasis may be a common mode of evolution, while modern data suggest very fast rates of evolution. One way to reconcile these observations is to imagine that evolution is punctuated, rather than gradual, on geological time scales. To test this hypothesis, we developed a novel maximum likelihood framework for fitting Lévy processes to comparative morphological data. This class of stochastic processes includes both a gradual and punctuated component. We found that a plurality of modern vertebrate clades examined are best fit by punctuated processes over models of gradual change, gradual stasis, and adaptive radiation. When we compare our results to theoretical expectations of the rate and speed of regime shifts for models that detail fitness landscape dynamics, we find that our quantitative results are broadly compatible with both microevolutionary models and with observations from the fossil record.A key debate in evolutionary biology centers around the seeming contradictions regarding the tempo and mode of evolution as seen in fossil data compared to ecological data. Fossil data often support models of stasis, in which little evolutionary change is seen within lineages over long timescales (Hunt 2007;Estes and Arnold 2007), while ecological data show that rapid bursts of evolution are not only possible, but potentially common (Reznick et al. 1997;Herrel et al. 2008;Brown and Brown 2013;Cresko et al. 2004). At face value, these observations seem to contradict one another, an observation known as the "paradox of stasis" (Hansen and Houle 2004). These observations are often reconciled through a descriptive model of punctuated evolution, entailing stasis interrupted by pulses of rapid change, as famously articulated by Simpson (Simpson 1944;1953).On macroevolutionary timescales, pulses of rapid change are expected to look roughly instantaneous. Only recently have statistical methods grown sophisticated enough to model punctuated evolution as a stochastic process, with advances showing that punctuation is detectable in some fossil time series (Hunt et al. 2015) and between pairs of living and extinct taxa (Uyeda et al. 2011).While these studies establish the existence of punctuated evolution, it is still unknown whether the evolutionary mode is common or rare. How many clades in the Tree of Life were shaped by abrupt pulses of rapid evolution? If these evolutionary pulses are common, then that should inform our expectations about how traits evolved for clades that left no fossils, and the potential for vulnerable species to adapt rapidly to climate change (Quintero and Wiens 2013). To this end, phylogenetic models-models of trait evolution that account for the shared ancestry of species-have played a vital role in measuring the relative support of competing Simpsonian modes of evolution.A pioneering meta-analysis (Harmon et al. 2010a) fitted a collection of phylogen...

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“…As observed in Kappus and Reiss (2010), identification of the Lévy measure from increments observed at arbitrary discrete grid is not trivial. In yuima, it is possible to estimate the parameters of a compound Poisson and a variance gamma process.…”

Section: It Is Worth Mentioning That the Increments G (R)mentioning

confidence: 99%

COGARCH (p,q) : Simulation and Inference with the yuima Package

Iacus¹,

Mercuri²,

Rroji³

2017

J. Stat. Soft.

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In this paper we show how to simulate and estimate a COGARCH(p, q) model in the R package yuima. Several routines for simulation and estimation are introduced. In particular, for the generation of a COGARCH(p, q) trajectory, the user can choose between two alternative schemes. The first is based on the Euler discretization of the stochastic differential equations that identify a COGARCH(p, q) model while the second considers the explicit solution of the equations defining the variance process.Estimation is based on the matching of the empirical with the theoretical autocorrelation function. Three different approaches are implemented: minimization of the mean squared error, minimization of the absolute mean error and the generalized method of moments where the weighting matrix is continuously updated. Numerical examples are given in order to explain methods and classes used in the yuima package.

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Estimation of the characteristics of a Lévy process observed at arbitrary frequency

Cited by 33 publications

References 15 publications

High-frequency Donsker theorems for Lévy measures

High-frequency Donsker theorems for Lévy measures

Punctuated evolution shaped modern vertebrate diversity

COGARCH (p,q) : Simulation and Inference with the yuima Package

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