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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  • Alfonsi, A. (2010). Cox–Ingersoll–Ross (CIR) model. In Encyclopedia of Quantitative Finance, ed. R. Cont. John Wiley, Chichester, pp. 401–403.
  • Chan, K. C., Karolyi, G. A., Longstaff, F. A. and Sanders, A. B. (1992). An empirical comparison of alternative models of the short-term interest rate. J. Finance 47, 1209–1227.
  • Comtet, L. (1974). Advanced Combinatorics: The Art of Finite and Infinite Expansions. Reidel, Dordrecht.
  • Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman & Hall/CRC, Boca Raton, FL.
  • Cox, J. C., Ingersoll, J. E., Jr. and Ross, S. A. (1985). A theory of the term structure of interest rates. Econometrica 53, 385–407.
  • Feller, W. (1951). Two singular diffusion problems. Ann. Math. (2) 54, 173–182.
  • Gordy, M. B. (2012). On the distribution of a discrete sample path of a square-root diffusion. Res. Rep. 2012-12, Federal Reserve Board.
  • Horn, R. A. and Johnson, C. R. (1985). Matrix Analysis. Cambridge University Press.
  • Jensen, D. R. (1969). Limit properties of noncentral multivariate Rayleigh and chi-square distributions. SIAM J. Appl. Math. 17, 802–814.
  • Jiang, G. J. and Knight, J. L. (2002). Estimation of continuous-time processes via the empirical characteristic function. J. Bus. Econom. Statist. 20, 198–212.
  • Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions, Vol. 1, 2nd edn. John Wiley, New York.
  • Kotz, S., Balakrishnan, N. and Johnson, N. L. (2000). Continuous Multivariate Distributions, Vol. 1, Models and Applications, 2nd edn. John Wiley, New York.
  • Krishnamoorthy, A. S. and Parthasarathy, M. (1951). A multivariate gamma-type distribution. Ann. Math. Statist. 22, 549–557. (Correction: 31 (1960), p. 229.)
  • Roy, S. N., Greenberg, B. G. and Sarhan, A. E. (1960). Evaluation of determinants, characteristic equations and their roots for a class of patterned matrices. J. R. Statist. Soc. B 22, 348–359.
  • Royen, T. (1994). On some multivariate gamma-distributions connected with spanning trees. Ann. Inst. Statist. Math. 46, 361–371.
  • Vandebril, R., Van Barel, M. and Mastronardi, N. (2008). Matrix Computations and Semiseparable Matrices, Vol. 1, Linear Systems. Johns Hopkins University Press, Baltimore, MD. \endharvreferences