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