On infinitely divisible multivariate gamma distributions

2021 ◽  
pp. 1-7
Author(s):  
Stephen G. Walker
Keyword(s):  
Infinitely Divisible ◽  
Gamma Distributions

scholarly journals Multifractality of products of geometric Ornstein-Uhlenbeck-type processes

2008 ◽  
Vol 40 (4) ◽  
pp. 1129-1156 ◽  
Author(s):  
V. V. Anh ◽  
Nikolai N. Leonenko ◽  
Narn-Rueih Shieh
Keyword(s):  
Dependence Structure ◽  
Inverse Gaussian ◽  
Infinitely Divisible ◽  
Covariance Functions ◽  
Lévy Motion ◽  
Gamma Distributions ◽  
Cumulative Processes ◽  
The Moment ◽  
Normal Inverse Gaussian ◽  
Mean Variance

We investigate the properties of multifractal products of geometric Ornstein-Uhlenbeck (OU) processes driven by Lévy motion. The conditions on the mean, variance, and covariance functions of the resulting cumulative processes are interpreted in terms of the moment generating functions. We consider five cases of infinitely divisible distributions for the background driving Lévy processes, namely, the gamma and variance gamma distributions, the inverse Gaussian and normal inverse Gaussian distributions, and the z-distributions. We establish the corresponding scenarios for the limiting processes, including their Rényi functions and dependence structure.

scholarly journals Infinitely divisible multivariate and matrix Gamma distributions

2014 ◽  
Vol 130 ◽  
pp. 155-175 ◽  
Author(s):  
Victor Pérez-Abreu ◽  
Robert Stelzer
Keyword(s):  
Infinitely Divisible ◽  
Gamma Distributions

scholarly journals Multifractality of products of geometric Ornstein-Uhlenbeck-type processes

2008 ◽  
Vol 40 (04) ◽  
pp. 1129-1156 ◽  
Author(s):  
V. V. Anh ◽  
Nikolai N. Leonenko ◽  
Narn-Rueih Shieh
Keyword(s):  
Dependence Structure ◽  
Inverse Gaussian ◽  
Infinitely Divisible ◽  
Covariance Functions ◽  
Lévy Motion ◽  
Gamma Distributions ◽  
Cumulative Processes ◽  
The Moment ◽  
Normal Inverse Gaussian ◽  
Mean Variance

We investigate the properties of multifractal products of geometric Ornstein-Uhlenbeck (OU) processes driven by Lévy motion. The conditions on the mean, variance, and covariance functions of the resulting cumulative processes are interpreted in terms of the moment generating functions. We consider five cases of infinitely divisible distributions for the background driving Lévy processes, namely, the gamma and variance gamma distributions, the inverse Gaussian and normal inverse Gaussian distributions, and the z-distributions. We establish the corresponding scenarios for the limiting processes, including their Rényi functions and dependence structure.

scholarly journals Non-Debye Relaxations: Two Types of Memories and Their Stieltjes Character

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 477
Author(s):  
Katarzyna Górska ◽  
Andrzej Horzela
Keyword(s):  
Stochastic Processes ◽  
Spectral Function ◽  
Evolution Equations ◽  
Memory Function ◽  
Relaxation Phenomena ◽  
Stieltjes Function ◽  
Infinitely Divisible ◽  
Spectral Functions ◽  
Debye Relaxation ◽  
Relaxation Functions

In this paper, we show that spectral functions relevant for commonly used models of the non-Debye relaxation are related to the Stieltjes functions supported on the positive semi-axis. Using only this property, it can be shown that the response and relaxation functions are non-negative. They are connected to each other and obey the time evolution provided by integral equations involving the memory function M(t), which is the Stieltjes function as well. This fact is also due to the Stieltjes character of the spectral function. Stochastic processes-based approach to the relaxation phenomena gives the possibility to identify the memory function M(t) with the Laplace (Lévy) exponent of some infinitely divisible stochastic processes and to introduce its partner memory k(t). Both memories are related by the Sonine equation and lead to equivalent evolution equations which may be freely interchanged in dependence of our knowledge on memories governing the process.

scholarly journals Zero-inflated-censored Weibull and gamma regression models to estimate wild boar population dispersal distance

2021 ◽  
Author(s):  
Eduardo de Freitas Costa ◽  
Silvana Schneider ◽  
Giulia Bagatini Carlotto ◽  
Tainá Cabalheiro ◽  
Mauro Ribeiro de Oliveira Júnior
Keyword(s):  
Wild Boar ◽  
Regression Models ◽  
Likelihood Function ◽  
Complex Structure ◽  
Dispersal Distance ◽  
Wild Boar Population ◽  
Gamma Distributions ◽  
Excess Of Zeros ◽  
Censored Observations ◽  
Positive Effect

AbstractThe dynamics of the wild boar population has become a pressing issue not only for ecological purposes, but also for agricultural and livestock production. The data related to the wild boar dispersal distance can have a complex structure, including excess of zeros and right-censored observations, thus being challenging for modeling. In this sense, we propose two different zero-inflated-right-censored regression models, assuming Weibull and gamma distributions. First, we present the construction of the likelihood function, and then, we apply both models to simulated datasets, demonstrating that both regression models behave well. The simulation results point to the consistency and asymptotic unbiasedness of the developed methods. Afterwards, we adjusted both models to a simulated dataset of wild boar dispersal, including excess of zeros, right-censored observations, and two covariates: age and sex. We showed that the models were useful to extract inferences about the wild boar dispersal, correctly describing the data mimicking a situation where males disperse more than females, and age has a positive effect on the dispersal of the wild boars. These results are useful to overcome some limitations regarding inferences in zero-inflated-right-censored datasets, especially concerning the wild boar’s population. Users will be provided with an R function to run the proposed models.

scholarly journals Survival and Reliability Analysis with an Epsilon-Positive Family of Distributions with Applications

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 908
Author(s):  
Perla Celis ◽  
Rolando de la Cruz ◽  
Claudio Fuentes ◽  
Héctor W. Gómez
Keyword(s):  
Survival Data ◽  
Maximum Likelihood Estimates ◽  
New Class ◽  
New Family ◽  
Gamma Distributions ◽  
Special Cases ◽  
Log Normal ◽  
Positive Support ◽  
Positive Distributions ◽  
Family Of Distributions

We introduce a new class of distributions called the epsilon–positive family, which can be viewed as generalization of the distributions with positive support. The construction of the epsilon–positive family is motivated by the ideas behind the generation of skew distributions using symmetric kernels. This new class of distributions has as special cases the exponential, Weibull, log–normal, log–logistic and gamma distributions, and it provides an alternative for analyzing reliability and survival data. An interesting feature of the epsilon–positive family is that it can viewed as a finite scale mixture of positive distributions, facilitating the derivation and implementation of EM–type algorithms to obtain maximum likelihood estimates (MLE) with (un)censored data. We illustrate the flexibility of this family to analyze censored and uncensored data using two real examples. One of them was previously discussed in the literature; the second one consists of a new application to model recidivism data of a group of inmates released from the Chilean prisons during 2007. The results show that this new family of distributions has a better performance fitting the data than some common alternatives such as the exponential distribution.

Learning for infinitely divisible GARCH models in option pricing

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Fumin Zhu ◽  
Michele Leonardo Bianchi ◽  
Young Shin Kim ◽  
Frank J. Fabozzi ◽  
Hengyu Wu
Keyword(s):  
Option Pricing ◽  
Stock Returns ◽  
Bayesian Learning ◽  
Space Structure ◽  
Estimation Methods ◽  
Garch Models ◽  
Learning Approaches ◽  
Infinitely Divisible ◽  
Asymmetric Garch ◽  
Non Gaussian

AbstractThis paper studies the option valuation problem of non-Gaussian and asymmetric GARCH models from a state-space structure perspective. Assuming innovations following an infinitely divisible distribution, we apply different estimation methods including filtering and learning approaches. We then investigate the performance in pricing S&P 500 index short-term options after obtaining a proper change of measure. We find that the sequential Bayesian learning approach (SBLA) significantly and robustly decreases the option pricing errors. Our theoretical and empirical findings also suggest that, when stock returns are non-Gaussian distributed, their innovations under the risk-neutral measure may present more non-normality, exhibit higher volatility, and have a stronger leverage effect than under the physical measure.

Tail asymptotics of an infinitely divisible space-time model with convolution equivalent Lévy measure

2021 ◽  
Vol 58 (1) ◽  
pp. 42-67 ◽  
Author(s):  
Mads Stehr ◽  
Anders Rønn-Nielsen
Keyword(s):  
Space Time ◽  
Time Interval ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Time Model ◽  
Spatial Object ◽  
Tail Behaviour ◽  
Fixed Radius ◽  
The Right ◽  
Space Time Model

AbstractWe consider a space-time random field on ${{\mathbb{R}^d} \times {\mathbb{R}}}$ given as an integral of a kernel function with respect to a Lévy basis with a convolution equivalent Lévy measure. The field obeys causality in time and is thereby not continuous along the time axis. For a large class of such random fields we study the tail behaviour of certain functionals of the field. It turns out that the tail is asymptotically equivalent to the right tail of the underlying Lévy measure. Particular examples are the asymptotic probability that there is a time point and a rotation of a spatial object with fixed radius, in which the field exceeds the level x, and that there is a time interval and a rotation of a spatial object with fixed radius, in which the average of the field exceeds the level x.

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