**Rational Approximation of Transfer Functions for Non-Negative EPT Densities**

**Authors: Conor Sexton, Martine Olivi & Bernard Hanzon**

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A stable Exponential-Polynomial-Trigonometric (EPT) probability density function is fitted to a large set of financial data. The class of EPT functions have a strictly proper continuous time rational transform. An isometry is used to derive its discrete time transfer function, also rational. This function can be written in terms of Fourier coefficients which are used as inputs to RARL2. A minimal state space realization is returned to approximate the density function. Non-negativity of the EPT function must be examined which is carried out using the Budan-Fourier technique. A convex optimisation algorithm is then implemented to ensure an optimal non-negative approximation.