Journal of Applied Probability

Finite-dimensional distributions of a square-root diffusion

Michael B. Gordy

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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.

Article information

J. Appl. Probab. Volume 51, Number 4 (2014), 930-942.

First available in Project Euclid: 20 January 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G17: Sample path properties
Secondary: 60E10: Characteristic functions; other transforms

Bell polynomial CIR process difference of gamma variates Kibble-Moran distribution Krishnamoorthy-Parthasarathy distribution multivariate noncentral chi-squared distribution multivariate gamma distribution square-root diffusion


Gordy, Michael B. Finite-dimensional distributions of a square-root diffusion. J. Appl. Probab. 51 (2014), no. 4, 930--942.

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