Representation of Non-Gaussian Continuous Random Variables by a Taylor Series in Standard Gaussian Random Variable: Estimation, Analytics and Applications

Abstract

We give a series representation of non-Gaussian continuous random variables in terms of a Taylor Series (Z-Series) in Standard Gaussian random variable or alternatively a series in Hermite polynomials (Hermite-Series) of Standard Gaussian random variable. We present simple numerical methods to find coefficients of these series given the density, data or moments of the non-Gaussian random variable. We also show how to do basic operations like addition, subtraction, and taking functions of non-Gaussian random variables using their Z-series. We present the concept of correlations between Hermite polynomials of Hermite-Series of two non-Gaussian random variables. These Hermite polynomial correlations are far more robust than the correlations between non-Gaussian random variables we presently see in literature. Then we present methods to calculate conditional expected values and distributions of a non-Gaussian random variable given the value or distribution of another correlated non-Gaussian random variable. We apply the conditional distribution knowledge to calculation of regression coefficients in a Hermite-Series regression. At the end, We apply the methods learnt to various applications in simulation of Stochastic differential equations, financial trading and hedging of securities.

Keywords: Non-Gaussian Continuous Random variables, Non-Gaussian Conditional Expectation, Correlations, and Hermite-Series Regression