A Stochastic Differential Equation for Modeling the “Classical” Probability Distributions

Stochastic differential equations are a flexible way to model continuous probability distributions. The most popular differential equations are for non-stationary Lognormal, non-stationary Normal and stationary Ornstein-Uhlenbeck distributions. The probability densities are known for these distributions and the assumptions behind the differential equations are well understood. Unfortunately, the assumptions do not fit most situations. In economics and finance, prices and quantities are usually stationary and positive. The Lognormal and Normal distributions are nonstationary and the Normal and Ornstein-Uhlenbeck distributions allow negative prices and quantities. This study derives a stochastic differential equation that includes most of the classical probability distributions as special cases and greatly expands the number distributions that can be used in models of stochastic dynamic systems.