We derive multivariate moment generating functions for the conditional and stationary distributions of a discrete sample path of n observations of a square-root diffusion (CIR) process, X(t). For any fixed vector of observation times t1,…,tn, we find the conditional joint distribution of (X(t1),…,X(tn)) is a multivariate noncentral chi-squared distribution and the stationary joint distribution is a Krishnamoorthy-Parthasarathy multivariate gamma distribution. Multivariate cumulants of the stationary distribution have a simple and computationally tractable expression. We also obtain the moment generating function for the increment X(t + δ) - X(t), and show that the increment is equivalent in distribution to a scaled difference of two independent draws from a gamma distribution.
On the Distribution of a Discrete Sample Path of a Square-Root Diffusion