Levy Processes and Infinitely Divisible Distributions

2001 ◽  
Vol 85 (504) ◽  
pp. 568 ◽  
Author(s):  
David Applebaum ◽  
Ken-iti Sato
Keyword(s):  
Lévy Processes ◽  
Levy Processes ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible

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.

FRACTIONAL LÉVY PROCESSES AND NOISES ON GEL′FAND TRIPLE

2010 ◽  
Vol 10 (01) ◽  
pp. 37-51 ◽  
Author(s):  
ZHIYUAN HUANG ◽  
XUEBIN LÜ ◽  
JIANPING WAN
Keyword(s):  
Integral Method ◽  
Lévy Processes ◽  
Fractional Integral ◽  
Levy Processes ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
White Noises

In this paper, we construct a class of infinitely divisible distributions on Gel′fand triple. Based on this construction, we define Lévy processes on Gel′fand triple and give their Lévy–Itô decompositions. Then, we construct the general Lévy white noises on Gel′fand triple. By using the Riemann–Liouville fractional integral method, we define the general fractional Lévy noises on Gel′fand triple and investigate their distribution properties.

An operational calculus for probability distributions via Laplace transforms

1996 ◽  
Vol 28 (01) ◽  
pp. 75-113 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
Lévy Processes ◽  
Probability Distributions ◽  
Transfer Operator ◽  
Operational Calculus ◽  
Levy Processes ◽  
Infinitely Divisible Distributions ◽  
Service Time Distribution ◽  
Positive Real ◽  
Infinitely Divisible ◽  
Special Case

In this paper we investigate operators that map one or more probability distributions on the positive real line into another via their Laplace–Stieltjes transforms. Our goal is to make it easier to construct new transforms by manipulating known transforms. We envision the results here assisting modelling in conjunction with numerical transform inversion software. We primarily focus on operators related to infinitely divisible distributions and Lévy processes, drawing upon Feller (1971). We give many concrete examples of infinitely divisible distributions. We consider a cumulant-moment-transfer operator that allows us to relate the cumulants of one distribution to the moments of another. We consider a power-mixture operator corresponding to an independently stopped Lévy process. The special case of exponential power mixtures is a continuous analog of geometric random sums. We introduce a further special case which is remarkably tractable, exponential mixtures of inverse Gaussian distributions (EMIGs). EMIGs arise naturally as approximations for busy periods in queues. We show that the steady-state waiting time in an M/G/1 queue is the difference of two EMIGs when the service-time distribution is an EMIG. We consider several transforms related to first-passage times, e.g. for the M/M/1 queue, reflected Brownian motion and Lévy processes. Some of the associated probability density functions involve Bessel functions and theta functions. We describe properties of the operators, including how they transform moments.

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.

An operational calculus for probability distributions via Laplace transforms

1996 ◽  
Vol 28 (1) ◽  
pp. 75-113 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
Lévy Processes ◽  
Probability Distributions ◽  
Transfer Operator ◽  
Operational Calculus ◽  
Levy Processes ◽  
Infinitely Divisible Distributions ◽  
Service Time Distribution ◽  
Positive Real ◽  
Infinitely Divisible ◽  
Special Case

In this paper we investigate operators that map one or more probability distributions on the positive real line into another via their Laplace–Stieltjes transforms. Our goal is to make it easier to construct new transforms by manipulating known transforms. We envision the results here assisting modelling in conjunction with numerical transform inversion software. We primarily focus on operators related to infinitely divisible distributions and Lévy processes, drawing upon Feller (1971). We give many concrete examples of infinitely divisible distributions. We consider a cumulant-moment-transfer operator that allows us to relate the cumulants of one distribution to the moments of another. We consider a power-mixture operator corresponding to an independently stopped Lévy process. The special case of exponential power mixtures is a continuous analog of geometric random sums. We introduce a further special case which is remarkably tractable, exponential mixtures of inverse Gaussian distributions (EMIGs). EMIGs arise naturally as approximations for busy periods in queues. We show that the steady-state waiting time in an M/G/1 queue is the difference of two EMIGs when the service-time distribution is an EMIG. We consider several transforms related to first-passage times, e.g. for the M/M/1 queue, reflected Brownian motion and Lévy processes. Some of the associated probability density functions involve Bessel functions and theta functions. We describe properties of the operators, including how they transform moments.

scholarly journals Asymmetrically tempered stable distributions with applications to finance

2019 ◽  
Vol 39 (1) ◽  
pp. 85-98
Author(s):  
A. Arefi ◽  
R. Pourtaheri
Keyword(s):  
Lévy Processes ◽  
Stable Distributions ◽  
Levy Processes ◽  
Infinitely Divisible ◽  
Financial Modeling ◽  
New Family ◽  
Tempered Stable Distributions ◽  
Exponential Lévy Models

In this paper, we introduce a technique to produce a new family of tempered stable distributions. We call this family asymmetrically tempered stable distributions.We provide two examples of this family named asymmetrically classical modified tempered stable ACMTS and asymmetrically modified classical tempered stable AMCTS distributions. Since the tempered stable distributions are infinitely divisible, Levy processes can be induced by the ACMTS and AMCTS distributions. The properties of these distributions will be discussed along with the advantages in applying them to financial modeling. Furthermore, we develop exponential Levy models for them. To demonstrate the advantages of the exponential Levy ACMTS and AMCTS models, we estimate parameters for the S&P 500 Index.

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