2-EPT probability density function: Difference between revisions

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{{AFC submission|t|reason|3=Submission needs more [[WP:RS|reliable sources]]. |ts=20120122123005|u=Hcsexton|ns=5}}

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{{Probability distribution |

name =2-EPT Density Function|

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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> <br/> <br/>

<math>\mathbbmathfrak{RRe}(\sigma(\textbf{A}_P))<0</math><br/> <br/>

<math>\mathbbmathfrak{RRe}(\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 & \mboxtext{if }x < 0

\\[8pt]

\textbf{c}_Pe^{\textbf{A}_Px}\textbf{b}_P & \mboxtext{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 & \mboxtext{if }x < 0

\\[8pt]

1 + \textbf{c}_P\textbf{A}_P^{-1}e^{\textbf{A}_Px}\textbf{b}_P & \mboxtext{if }x \geq 0

\end{matrix}\right.

</math>|

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}}

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).

The class of probability density functions on <math>\mathbb{R}</math> with strictly proper rational [[Characteristic function (probability theory)|characteristic function]] are referred to as 2-EPT probability density functions. On <math>[0, \infty)</math> as well as <math>(-\infty, 0)</math> these probability density functions are Exponential-Polynomial-Trigonometric (EPT) functions. An 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>. 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>. We have that <math>(\textbf{A}_N,\textbf{A}_P)</math> are square matrices, <math>(\textbf{b}_N,\textbf{b}_P)</math> column vectors and <math>(\textbf{c}_N,\textbf{c}_P)</math> row vectors. <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., 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 is shown that the class of 2-EPT densities is closed under many operations and using minimal realizations these calculations are illustrated for the two-sided framework in Sexton and Hanzon<ref>C. Sexton and B. Hanzon: 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 the open left and resp. open right half plane. The [[variance-gamma distribution]] density is shown to be a 2-EPT density under a parameter restriction and the [[Variance gamma process]]<ref>Madan, D., Carr, P., Chang, E., "The Variance Gamma Process and Option Pricing",European Finance Review 2: 79–105, 1998.</ref> can be implemented to demonstrate the benefits of adopting such an approach for financial modelling purposes. Examples of applications provided include option pricing, computing the Greeks and risk management calculations.

==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 class of probability density functions on <math>\mathbb{R}</math> with strictly proper rational [[Characteristic function (probability theory)|characteristic function]] are referred to as 2-EPT probability density functions. On <math>[0, \infty)</math> as well as <math>(-\infty, 0)</math> these probability density functions are Exponential-Polynomial-Trigonometric (EPT) functions. An 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>. 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>. We have that <math>(\textbf{A}_N,\textbf{A}_P)</math> are square matrices, <math>(\textbf{b}_N,\textbf{b}_P)</math> column vectors and <math>(\textbf{c}_N,\textbf{c}_P)</math> row vectors. <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., 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|Phasetype Typedistribution|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 ishas been shown that the class of 2-EPT densities is closed under many operations and using minimal realizations these calculations arehave been illustrated for the two-sided framework in Sexton and Hanzon.<ref>Sexton, C. Sexton and Hanzon, B., Hanzon: "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 and resp.or open right half plane. The [[variance-gamma distribution]] density ishas been shown to be a 2-EPT density under a parameter restriction and the [[Variance gamma process]].<ref>Madan, D., Carr, P., Chang, E., (1998) "The Variance Gamma Process and Option Pricing", ''European Finance Review'' 2: 79–105, 1998.</ref> can be implemented to demonstrate the benefits of adopting such an approach for financial modelling purposes. Examples of applications provided include option pricing, computing the Greeks and risk management calculations.

== 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:ContinuousTypes of probability distributions]]

{{statistics-stub}}

[[ru:Распределение variance-gamma]]