**State Space Calculations for two-sided EPT Densities with Financial Modelling Applications**

The class of probability density functions on R with strictly proper rational characteristic functions are considered. On [0, ∞) as well as (-∞, 0) these probability density functions are Exponential-Polynomial-Trigonometric (EPT) functions which are abbreviated as 2-EPT densities. EPT density functions can be represented as f(x)=

**c**e^{Ax}**b**, where**A**is a square matrix,**b**a column vector and**c**a row vector. The triple (**A, b, c**) is the realization of the EPT function. The general class of probability measures on R with (proper) rational characteristic functions is also considered whose densities correspond to mixtures of the pointmass at zero ("delta distribution") and 2-EPT densities. Unlike phase-type and matrix analytic 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 we illustrate how these calculations are carried out in this two-sided framework. The most significant contribution of the paper 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 density is shown to be a 2-EPT density under a parameter restriction and the Variance Gamma asset price process is 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.