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\textbf{c}_Ne^{\textbf{A}_Nx}\textbf{b}_N & \text{if }x < 0
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\textbf{c}_Pe^{\textbf{A}_Px}\textbf{b}_P & \text{if }x \geq 0
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In [[probability theory]], a '''2-EPT probability density function''' is a class of [[probability density function]]s on
==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.
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>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 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. 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, "www.2-ept.com"</ref>. It can be shown using Parsevals Theorem and a isometry that approximating the discrete time rational transform is equivalent to approximating the 2-EPT density itslef 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., "Parametrization of Rational Lossless Matrices with Applications to Linear System Theory", HDR Thesis, 2010</ref>. ▼
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 (
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 itslef 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}}</ref>.
==Applications==
Examples of applications include option pricing, computing the Greeks and risk management calculations.{{cn}} 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", "www.2-ept.com"</ref>
== Notes ==
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