Board of Governors of the Federal Reserve System

March 2012

On the distribution of a discrete sample path of a square-root diffusion

Michael B. Gordy

Abstract:

We derive the multivariate moment generating function (mgf) for the stationary distribution of a discrete sample path of n observations of a square-root diffusion (CIR) process, X(t). The form of the mgf establishes that the stationary joint distribution of (X(t(1)),...,X(t(n))) for any fixed vector of observation times (t(1),...,t(n)) is a Krishnamoorthy-Parthasarathy multivariate gamma distribution. As a corollary, we obtain the mgf for the increment X(t+dt)-X(t), and show that the increment is equivalent in distribution to a scaled difference of two independent draws from a gamma distribution. Simple closed-form solutions for the moments of the increments are given.

Keywords: Square-root diffusion, CIR process, multivariate gamma distribution, difference of gamma variates, Krishnamoorthy-Parthasarathy distribution, Kibble-Moran distribution, Bell polynomials

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