On the Distribution of a Discrete Sample Path of a Square-Root Diffusion

12 Pages Posted: 4 May 2012

Date Written: March 1, 2012


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

JEL Classification: C46, C58

Suggested Citation

Gordy, Michael B., On the Distribution of a Discrete Sample Path of a Square-Root Diffusion (March 1, 2012). FEDS Working Paper No. 2012-12. Available at SSRN: https://ssrn.com/abstract=2051017 or http://dx.doi.org/10.2139/ssrn.2051017

Michael B. Gordy (Contact Author)

Federal Reserve Board ( email )

20th & C. St., N.W.
Washington, DC 20551
United States
202-452-3705 (Phone)

HOME PAGE: http://https://www.federalreserve.gov/econres/michael-b-gordy.htm

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