Diffusion Processes with Polynomial Eigenfunctions

The aim of this paper is to characterize the one-dimensional stochastic differential equations, for which the eigenfunctions of the infinitesimal generator are polynomials in y. Affine transformations of the Ornstein-Uhlenbeck process, the Cox-Ingersoll-Ross process and the Jacobi process belong to the solutions of this stochastic differential equation family. Such processes exhibit specific patterns of the drift and volatility functions and can be represented by means of a basis of polynomial transforms which can be used to approximate the likelihood function. We also discuss the constraints on parameters to ensure the nonnegativity of the volatility function and the stationarity of the process. The possibility to fully characterize the dynamic properties of these processes explain why they are benchmark models for unconstrained variables such as asset returns (Ornstein-Uhlenbeck), for nonnegative variables as volatilities or interest rates (Cox, Ingersoll, Ross), or for variables which can be interpreted as probabilities (Jacobi).