View source for 2-EPT probability density function

{{Probability distribution |
  name       =2-EPT Density Function|
  type       =density|
  pdf_image  =|
  cdf_image  =|
  parameters = <math>(\textbf{A}_N,\textbf{b}_N,\textbf{c}_N,\textbf{A}_P,\textbf{b}_P,\textbf{c}_P)</math>

<math>\mathfrak{Re}(\sigma(\textbf{A}_P))<0</math>

<math>\mathfrak{Re}(\sigma(\textbf{A}_N))>0</math>|
  support    =<math>x \in (-\infty; +\infty)\!</math>|
  pdf        = <math>f(x) = 
    \left\{\begin{matrix}
      \textbf{c}_Ne^{\textbf{A}_Nx}\textbf{b}_N & \text{if }x < 0
      \\[8pt]
      \textbf{c}_Pe^{\textbf{A}_Px}\textbf{b}_P & \text{if }x \geq 0
    \end{matrix}\right.
  </math>|
  cdf        = <math>F(x) = 
    \left\{\begin{matrix}
      \textbf{c}_N\textbf{A}_N^{-1}e^{\textbf{A}_Nx}\textbf{b}_N & \text{if }x < 0
      \\[8pt]
      1 + \textbf{c}_P\textbf{A}_P^{-1}e^{\textbf{A}_Px}\textbf{b}_P & \text{if }x \geq 0
    \end{matrix}\right.
  </math>|
  mean       =<math> -\textbf{c}_N(-\textbf{A}_N)^{-2}\textbf{b}_N + \textbf{c}_P(-\textbf{A}_P)^{-2}\textbf{b}_P</math>|
  median     =|
  mode       =|
  variance   =|
  skewness   =<!-- to do -->||
  kurtosis   =<!-- to do -->||
  entropy    =<!-- to do -->|
  mgf        =|
  char       =<math> -\textbf{c}_N(Iiu-\textbf{A}_N)^{-1}\textbf{b}_N+\textbf{c}_P(Iiu-\textbf{A}_P)^{-1}\textbf{b}_P</math>|
}}

In [[probability theory]], a '''2-EPT probability density function''' is a class of [[probability density function]]s on the real line. The class contains the density functions of all distributions that have [[Characteristic function (probability theory)|characteristic function]]s that are strictly proper [[rational function]]s (i.e., the degree of the numerator is strictly less than the degree of the denominator).

==Definition==

A 2-EPT probability density function is a [[probability density function]] on <math>\mathbb{R}</math> with a strictly proper rational [[Characteristic function (probability theory)|characteristic function]].  On either <math>[0, +\infty)</math> or <math>(-\infty, 0)</math> these probability density functions are exponential-polynomial-trigonometric (EPT) functions.

Any EPT density function on <math>(-\infty, 0)</math> can be represented as
:<math>f(x)=\textbf{c}_Ne^{\textbf{A}_Nx}\textbf{b}_N ,</math>
where ''e'' represents a matrix exponential, <math>(\textbf{A}_N,\textbf{A}_P)</math> are square matrices, <math>(\textbf{b}_N,\textbf{b}_P)</math> are column vectors and <math>(\textbf{c}_N,\textbf{c}_P)</math> are row vectors.  Similarly the EPT density function on <math>[0, -\infty)</math> is expressed as 
:<math>f(x)=\textbf{c}_Pe^{\textbf{A}_Px}\textbf{b}_P.</math>

The parameterization <math>(\textbf{A}_N,\textbf{b}_N,\textbf{c}_N,\textbf{A}_P,\textbf{b}_P,\textbf{c}_P)</math>
is the minimal realization<ref>Kailath, T. (1980) ''Linear Systems'', Prentice Hall, 1980</ref> of the 2-EPT function.

The general class of probability measures on <math>\mathbb{R}</math> with (proper) rational characteristic functions are densities corresponding to mixtures of the pointmass at zero ("[[delta distribution]]") and 2-EPT densities. Unlike [[Phase-type distribution|phase-type]] and matrix geometric<ref>Neuts, M. "Probability Distributions of Phase Type", Liber Amicorum Prof. Emeritus H. Florin pages 173-206, Department of Mathematics, University of Louvain, Belgium  1975</ref> distributions, the 2-EPT probability density functions are defined on the whole real line. It has been shown that the class of 2-EPT densities is closed under many operations and using minimal realizations these calculations have been illustrated for the two-sided framework in Sexton and Hanzon.<ref>Sexton, C. and Hanzon, B., "State Space Calculations for two-sided EPT Densities with Financial Modelling Applications", ''www.2-ept.com''</ref> The most involved operation is the [[convolution]] of 2-EPT densities using state space techniques. Much of the work centers on the ability to decompose the rational characteristic function into the sum of two rational functions with poles located in either the open left or open right half plane. The [[variance-gamma distribution]] density has been shown to be a 2-EPT density under a parameter restriction.<ref>Madan, D., Carr, P., Chang, E. (1998) "The Variance Gamma Process and Option Pricing", ''European Finance Review'' 2: 79–105</ref>

== Notes ==
<references/>

==External links==
*[http://www.2-ept.com/ 2 - Exponential-Polynomial-Trigonometric (2-EPT) Probability Density Functions] {{Webarchive|url=https://web.archive.org/web/20200708015221/http://www.2-ept.com/ |date=2020-07-08 }} Website for background and Matlab implementations

{{ProbDistributions|continuous-infinite}}

{{DEFAULTSORT:Variance-Gamma Distribution}}
[[Category:Types of probability distributions]]

[[ru:Распределение variance-gamma]]