A Stochastic Diffusion Process for the Dirichlet Distribution

J. Bakosi and J. R. Ristorcelli

International Journal of Stochastic Analysis, 2013, vol. 2013, 1-7

Abstract:

The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability of N coupled stochastic variables with the Dirichlet distribution as its asymptotic solution. To ensure a bounded sample space, a coupled nonlinear diffusion process is required: the Wiener processes in the equivalent system of stochastic differential equations are multiplicative with coefficients dependent on all the stochastic variables. Individual samples of a discrete ensemble, obtained from the stochastic process, satisfy a unit-sum constraint at all times. The process may be used to represent realizations of a fluctuating ensemble of N variables subject to a conservation principle. Similar to the multivariate Wright-Fisher process, whose invariant is also Dirichlet, the univariate case yields a process whose invariant is the beta distribution. As a test of the results, Monte Carlo simulations are used to evolve numerical ensembles toward the invariant Dirichlet distribution.

Date: 2013
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnijsa:842981

DOI: 10.1155/2013/842981

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