A stochastic diffusion process for Lochner's generalized Dirichlet distribution

Abstract

The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability of N stochastic variables with Lochner's generalized Dirichlet distribution [R. H. Lochner, “A generalized Dirichlet distribution in Bayesian life testing,” J. R. Stat. Soc. Ser. B (Methodol.) 37(1), 103–113 (1975)] as its asymptotic solution. Individual samples of a discrete ensemble, obtained from the system of stochastic differential equations, equivalent to the Fokker-Planck equation developed here, satisfy a unit-sum constraint at all times and ensure a bounded sample space, similarly to the process developed in [J. Bakosi and J. R. Ristorcelli, “A stochastic diffusion process for the Dirichlet distribution,” Int. J. Stoch. Anal. 2013, 7]. Consequently, the generalized Dirichlet diffusion process may be used to represent realizations of a fluctuating ensemble of N variables subject to a conservation principle. Compared to the Dirichlet distribution and process, the additional parameters of the generlized Dirichlet distribution allow a more general class of physical processes to be modeled with a more general covariance matrix.