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Finite-Dimensional Distributions of a Square-Root Diffusion

Part of: Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Michael B. Gordy
Michael B. Gordy*
Affiliation:
Federal Reserve Board
*
∗ Postal address: Federal Reserve Board, Washington, DC, 20551, USA. Email address: [email protected]
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Abstract

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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.

Keywords

Bell polynomialCIR processdifference of gamma variatesKibble-Moran distributionKrishnamoorthy-Parthasarathy distributionmultivariate noncentral chi-squared distributionmultivariate gamma distributionsquare-root diffusion

MSC classification

Primary: 60G17: Sample path properties
Secondary: 60E10: Characteristic functions; other transforms
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 4 , December 2014 , pp. 930 - 942
Copyright
© Applied Probability Trust 

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