2-EPT probability density function: Difference between revisions

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{{Orphan|date=February 2012}}

{{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>

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

<math>\mathbbmathfrak{RRe}(\sigma(\textbf{A}_N))>0</math>|

support =<math>x \in (-\infty; +\infty)\!</math>|

pdf = <math>f(x) =

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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{{clarify|reason=link or explain for "strictly proper"|date=February 2012}} [[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

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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 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</ref> can be implemented to demonstrate the benefits of adopting such an approach for financial modelling purposes.

It can be shown using [[Parseval's theorem]] and an isometry that approximating the discrete time rational transform is equivalent to approximating the 2-EPT density itself in the L-2 Norm sense. The rational approximation software RARL2 is used to approximate the discrete time rational characteristic function of the density.<ref>Olivi, M. (2010) "Parametrization of Rational Lossless Matrices with Applications to Linear System Theory", HDR Thesis {{full|date=November 2012}}</ref>

==Applications==

Examples of applications include option pricing, computing the Greeks and risk management calculations.{{citation needed|date=February 2012}} Fitting 2-EPT density functions to empirical data has also been considered.<ref>Sexton, C., Olivi, M., Hanzon, B, "Rational Approximation of Transfer Functions for Non-Negative EPT Densities", [http://www.2-ept.com/uploads/8/2/3/7/8237897/rational_approximation_of_transfer_functions_for_non-negative_ept_densities.pdf Draft paper]</ref>

== Notes ==

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==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}}