scholarly journals A direct approach to the stable distributions

2016 ◽  
Vol 48 (A) ◽  
pp. 261-282 ◽  
Author(s):  
E. J. G. Pitman ◽  
Jim Pitman
Keyword(s):  
Functional Equation ◽  
Characteristic Function ◽  
Integral Representation ◽  
Explicit Form ◽  
Regular Variation ◽  
Stable Distribution ◽  
Stable Distributions ◽  
Direct Approach ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible

AbstractThe explicit form for the characteristic function of a stable distribution on the line is derived analytically by solving the associated functional equation and applying the theory of regular variation, without appeal to the general Lévy‒Khintchine integral representation of infinitely divisible distributions.

scholarly journals Conditional Characteristic Functions of Molchan-Golosov Fractional Lévy Processes with Application to Credit Risk

2013 ◽  
Vol 50 (4) ◽  
pp. 983-1005 ◽  
Author(s):  
Holger Fink
Keyword(s):  
Brownian Motion ◽  
Characteristic Function ◽  
Fractional Calculus ◽  
Hazard Rate ◽  
Lévy Processes ◽  
Levy Processes ◽  
Characteristic Functions ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Fractional Stochastic Differential Equations

Molchan-Golosov fractional Lévy processes (MG-FLPs) are introduced by way of a multivariate componentwise Molchan-Golosov transformation based on an n-dimensional driving Lévy process. Using results of fractional calculus and infinitely divisible distributions, we are able to calculate the conditional characteristic function of integrals driven by MG-FLPs. This leads to important predictions concerning multivariate fractional Brownian motion, fractional subordinators, and general fractional stochastic differential equations. Examples are the fractional Lévy Ornstein-Uhlenbeck and Cox-Ingersoll-Ross models. As an application we present a fractional credit model with a long range dependent hazard rate and calculate bond prices.

A Condition For Existence Of Moments Of Infinitely Divisible Distributions

1956 ◽  
Vol 8 ◽  
pp. 69-71 ◽  
Author(s):  
J. M. Shapiro
Keyword(s):  
Characteristic Function ◽  
Nondecreasing Function ◽  
Divisible Distribution ◽  
Infinitely Divisible Distributions ◽  
Real Constant ◽  
Infinitely Divisible Distribution ◽  
Infinitely Divisible ◽  
Image Position

Let F(x) be an infinitely divisible distribution and let ϕ(t) be its characteristic function. As is well known according to the formula of Lévy and Khintchine, ϕ(t) has the following representation:1where γ is a real constant and G(u) is a bounded nondecreasing function.

Singular infinitely divisible distributions whose characteristic functions vanish at infinity

1977 ◽  
Vol 82 (2) ◽  
pp. 277-287 ◽  
Author(s):  
Gavin Brown
Keyword(s):  
Dynamical Systems ◽  
Characteristic Function ◽  
Random Variables ◽  
Characteristic Functions ◽  
Strong Mixing ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Full Support ◽  
Singular Spectrum

In the course of discussing dynamical systems which enjoy strong mixing but have singular spectrum, E. Hewitt and the author, (2), recently constructed families of symmetric random variables which satisfy inter alia the following properties:(i) Zt is purely singular and has full support,(ii) χ(t)(x) → 0 as ± x → ∞, where χ(t) is the characteristic function of Zt,(iii)′ Zt+s, Zt + Zs have the same null events,(iv) whenever s ≠ t, Zt and Zs + a are mutually singular for every (possibly zero) constant a.

A functional equation and its application to the characterization of the Weibull and stable distributions

1976 ◽  
Vol 13 (2) ◽  
pp. 385-391 ◽  
Author(s):  
Y. H. Wang
Keyword(s):  
Functional Equation ◽  
Weibull Distribution ◽  
Exponential Distribution ◽  
Stable Distribution ◽  
Stable Distributions ◽  
Cauchy Functional Equation ◽  
Linear Statistics

The Cauchy functional equation Φ(x + y) = Φ(x) + Φ(y) is generalized to the form , assuming Φ is left- or right- continuous. This result is used to obtain (1) a characterization of the Weibull distribution, in the spirit of the memoryless property of the exponential distribution, by , for all x, y ≧ 0;(2) a characterization of the symmetric α-stable distribution by the equidistribution of linear statistics.

A functional equation and its application to the characterization of the Weibull and stable distributions

1976 ◽  
Vol 13 (02) ◽  
pp. 385-391
Author(s):  
Y. H. Wang
Keyword(s):  
Functional Equation ◽  
Weibull Distribution ◽  
Exponential Distribution ◽  
Stable Distribution ◽  
Stable Distributions ◽  
Cauchy Functional Equation ◽  
Linear Statistics

The Cauchy functional equation Φ(x+y) = Φ(x) + Φ(y) is generalized to the form, assuming Φ is left- or right- continuous. This result is used to obtain (1) a characterization of the Weibull distribution, in the spirit of the memoryless property of the exponential distribution, by, for allx,y≧ 0;(2) a characterization of the symmetricα-stable distribution by the equidistribution of linear statistics.

Conditional Characteristic Functions of Molchan-Golosov Fractional Lévy Processes with Application to Credit Risk

2013 ◽  
Vol 50 (04) ◽  
pp. 983-1005
Author(s):  
Holger Fink
Keyword(s):  
Brownian Motion ◽  
Characteristic Function ◽  
Fractional Calculus ◽  
Hazard Rate ◽  
Lévy Processes ◽  
Levy Processes ◽  
Characteristic Functions ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Fractional Stochastic Differential Equations

Molchan-Golosov fractional Lévy processes (MG-FLPs) are introduced by way of a multivariate componentwise Molchan-Golosov transformation based on ann-dimensional driving Lévy process. Using results of fractional calculus and infinitely divisible distributions, we are able to calculate the conditional characteristic function of integrals driven by MG-FLPs. This leads to important predictions concerning multivariate fractional Brownian motion, fractional subordinators, and general fractional stochastic differential equations. Examples are the fractional Lévy Ornstein-Uhlenbeck and Cox-Ingersoll-Ross models. As an application we present a fractional credit model with a long range dependent hazard rate and calculate bond prices.

scholarly journals On a new Class of Tempered Stable Distributions: Moments and Regular Variation

2012 ◽  
Vol 49 (4) ◽  
pp. 1015-1035 ◽  
Author(s):  
Michael Grabchak
Keyword(s):  
Regular Variation ◽  
Stable Distribution ◽  
Sufficient Conditions ◽  
Domain Of Attraction ◽  
Stable Distributions ◽  
Necessary And Sufficient Conditions ◽  
Tempered Stable Distribution ◽  
New Class ◽  
Tempered Stable Distributions ◽  
Necessary And Sufficient

We extend the class of tempered stable distributions, which were first introduced in Rosiński (2007). Our new class allows for more structure and more variety of the tail behaviors. We discuss various subclasses and the relations between them. To characterize the possible tails, we give detailed results about finiteness of various moments. We also give necessary and sufficient conditions for the tails to be regularly varying. This last part allows us to characterize the domain of attraction to which a particular tempered stable distribution belongs.

scholarly journals On a new Class of Tempered Stable Distributions: Moments and Regular Variation

2012 ◽  
Vol 49 (04) ◽  
pp. 1015-1035
Author(s):  
Michael Grabchak
Keyword(s):  
Regular Variation ◽  
Stable Distribution ◽  
Sufficient Conditions ◽  
Domain Of Attraction ◽  
Stable Distributions ◽  
Necessary And Sufficient Conditions ◽  
Tempered Stable Distribution ◽  
New Class ◽  
Tempered Stable Distributions ◽  
Necessary And Sufficient

We extend the class of tempered stable distributions, which were first introduced in Rosiński (2007). Our new class allows for more structure and more variety of the tail behaviors. We discuss various subclasses and the relations between them. To characterize the possible tails, we give detailed results about finiteness of various moments. We also give necessary and sufficient conditions for the tails to be regularly varying. This last part allows us to characterize the domain of attraction to which a particular tempered stable distribution belongs.

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