Annals of Statistics

Closed-form likelihood expansions for multivariate diffusions

Yacine Aït-Sahalia

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This paper provides closed-form expansions for the log-likelihood function of multivariate diffusions sampled at discrete time intervals. The coefficients of the expansion are calculated explicitly by exploiting the special structure afforded by the diffusion model. Examples of interest in financial statistics and Monte Carlo evidence are included, along with the convergence of the expansion to the true likelihood function.

Article information

Ann. Statist., Volume 36, Number 2 (2008), 906-937.

First available in Project Euclid: 13 March 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators 62M05: Markov processes: estimation
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65]

Diffusions likelihood expansions discrete observations


Aït-Sahalia, Yacine. Closed-form likelihood expansions for multivariate diffusions. Ann. Statist. 36 (2008), no. 2, 906--937. doi:10.1214/009053607000000622.

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  • Aït-Sahalia, Y. (1996). Testing continuous-time models of the spot interest rate. Rev. Financial Studies 9 385–426.
  • Aït-Sahalia, Y. (2002). Maximum-likelihood estimation of discretely-sampled diffusions: A closed-form approximation approach. Econometrica 70 223–262.
  • Aït-Sahalia, Y. and Kimmel, R. (2007). Maximum likelihood estimation of stochastic volatility models. J. Financial Economics 83 413–452.
  • Azencott, R. (1984). Densité des diffusions en temps petit: Développements asymptotiques. I. Seminar on Probability XVIII. Lecture Notes in Math. 1059 402–498. Springer, Berlin.
  • Bakshi, G. S. and Yu, N. (2005). A refinement to “Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach,” by Aït-Sahalia. J. Business 78 2037–2052.
  • Billingsley, P. (1961). Statistical Inference for Markov Processes. Univ. Chicago Press.
  • Black, F., Derman, E. and Toy, W. (1990). A one-factor model of interest rates and its applications to treasury bond options. Financial Analysts J. 46 33–39.
  • Cyganowski, S., Kloeden, P. and Ombach, J. (2001). From Elementary Probability to Stochastic Differential Equations with MAPLE. Springer. Berlin.
  • Dacunha-Castelle, D. and Florens-Zmirou, D. (1986). Estimation of the coefficients of a diffusion from discrete observations. Stochastics 19 263–284.
  • DiPietro, M. (2001). Bayesian inference for discretely sampled diffusion processes with financial applications. Ph.D. dissertation, Dept. Statistics, Carnegie Mellon Univ.
  • Doss, H. (1977). Lien entre equations différentielles stochastiques et ordinaires. Ann. Inst. H. Poincaré (B) 13 99–125.
  • Edwards, C. H. (1973). Advanced Calculus of Several Variables. Dover, New York.
  • Egorov, A. V., Li, H. and Xu, Y. (2003). Maximum likelihood estimation of time inhomogeneous diffusions. J. Econometrics 114 107–139.
  • Friedman, A. (1964). Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ.
  • Hansen, L. P. and Sargent, T. J. (1983). The dimensionality of the aliasing problem in models with rational spectral densities. Econometrica 51 377–387.
  • Heath, D., Jarrow, R. and Morton, A. (1992). Bond pricing and the term structure of interest rates: A new methodology for contingent claims evaluation. Econometrica 60 77–105.
  • Ho, T. S. Y. and Lee, S.-B. (1986). Term structure movements and pricing interest rate contingent claims. J. Finance 41 1011–1029.
  • Hull, J. and White, A. (1990). Pricing interest rate derivative securities. Rev. Financial Studies 3 573–592.
  • Hurn, A. S., Jeisman, J. and Lindsay, K. (2005). Seeing the wood for the trees: A critical evaluation of methods to estimate the parameters of stochastic differential equations. Technical report, School of Economics and Finance, Queensland Univ. Technology.
  • Jensen, B. and Poulsen, R. (2002). Transition densities of diffusion processes: Numerical comparison of approximation techniques. J. Derivatives 9 1–15.
  • Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus. Springer, New York.
  • Kessler, M. and Rahbek, A. (2004). Identification and inference for multivariate cointegrated and ergodic Gaussian diffusions. Statist. Inference Stochastic Process. 7 137–151.
  • McCullagh, P. (1987). Tensor Methods in Statistics. Chapman and Hall, London.
  • Philips, P. C. B. (1973). The problem of identification in finite parameter continuous time models. J. Econometrics 1 351–362.
  • Schaumburg, E. (2001). Maximum likelihood estimation of jump processes with applications to finance. Ph.D. thesis, Princeton Univ.
  • Stramer, O. and Yan, J. (2005). On simulated likelihood of discretely observed diffusion processes and comparison to closed-form approximation. Technical report, Univ. Iowa.
  • Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Springer, New York.
  • Sundaresan, S. M. (2000). Continuous-time finance: A review and an assessment. J. Finance 55 1569–1622.
  • Varadhan, S. R. S. (1967). Diffusion processes in a small time interval. Comm. Pure Appl. Math. 20 659–685.
  • Watanabe, S. and Yamada, T. (1971). On the uniqueness of solutions of stochastic differential equations. II. J. Math. Kyoto Univ. 11 553–563.
  • Withers, C. S. (2000). A simple expression for the multivariate Hermite polynomials. Statist. Probab. Lett. 47 165–169.
  • Yamada, T. and Watanabe, S. (1971). On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11 155–167.
  • Yu, J. (2003). Closed-form likelihood estimation of jump-diffusions with an application to the realignment risk premium of the Chinese Yuan. Ph.D. dissertation, Princeton Univ.