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
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