The Annals of Statistics

On the approximate maximum likelihood estimation for diffusion processes

Jinyuan Chang and Song Xi Chen

Full-text: Open access


The transition density of a diffusion process does not admit an explicit expression in general, which prevents the full maximum likelihood estimation (MLE) based on discretely observed sample paths. Aït-Sahalia [J. Finance 54 (1999) 1361–1395; Econometrica 70 (2002) 223–262] proposed asymptotic expansions to the transition densities of diffusion processes, which lead to an approximate maximum likelihood estimation (AMLE) for parameters. Built on Aït-Sahalia’s [Econometrica 70 (2002) 223–262; Ann. Statist. 36 (2008) 906–937] proposal and analysis on the AMLE, we establish the consistency and convergence rate of the AMLE, which reveal the roles played by the number of terms used in the asymptotic density expansions and the sampling interval between successive observations. We find conditions under which the AMLE has the same asymptotic distribution as that of the full MLE. A first order approximation to the Fisher information matrix is proposed.

Article information

Ann. Statist., Volume 39, Number 6 (2011), 2820-2851.

First available in Project Euclid: 24 January 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M05: Markov processes: estimation
Secondary: 62F12: Asymptotic properties of estimators

Asymptotic expansion asymptotic normality consistency discrete time observation maximum likelihood estimation


Chang, Jinyuan; Chen, Song Xi. On the approximate maximum likelihood estimation for diffusion processes. Ann. Statist. 39 (2011), no. 6, 2820--2851. doi:10.1214/11-AOS922.

Export citation


  • Aït-Sahalia, Y. (1999). Transition densities for interest rate and other nonlinear diffusions. J. Finance 54 1361–1395.
  • Aït-Sahalia, Y. (2002). Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach. Econometrica 70 223–262.
  • Aït-Sahalia, Y. (2008). Closed-form likelihood expansions for multivariate diffusions. Ann. Statist. 36 906–937.
  • Aït-Sahalia, Y. and Kimmel, R. (2007). Maximum likelihood estimation of stochastic volatility models. Journal of Financial Economics 83 413–452.
  • Aït-Sahalia, Y. and Kimmel, R. (2010). Estimating affine multifactor term structure models using closed-form likelihood expansions. Journal of Financial Economics 98 113–144.
  • Aït-Sahalia, Y. and Mykland, P. A. (2004). Estimators of diffusions with randomly spaced discrete observations: A general theory. Ann. Statist. 32 2186–2222.
  • Aït-Sahalia, Y., Mykland, P. A. and Zhang, L. (2011). Ultra high frequency volatility estimation with dependent microstructure noise. J. Econometrics 160 160–175.
  • Bakshi, G., Cao, C. and Chen, Z. (1997). Empirical performance of alternative option pricing models. J. Finance 52 2003–2049.
  • Beskos, A., Papaspiliopoulos, O., Roberts, G. O. and Fearnhead, P. (2006). Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 68 333–382.
  • Bibby, B. M. and Sørensen, M. (1995). Martingale estimation functions for discretely observed diffusion processes. Bernoulli 1 17–39.
  • Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York.
  • Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81 637–654.
  • Chang, J. and Chen, S. X. (2011). On the approximate maximum likelihood estimation for diffusion processes. Technical Report 2011-08, Center for Statistical Science, Peking Univ.
  • Cox, J. C., Ingersoll, J. E. and Ross, S. A. (1985). A theory of term structure of interest rates. Econometrica 53 385–407.
  • Cramér, H. (1946). Mathematical Methods of Statistics. Princeton Mathematical Series 9. Princeton Univ. Press, Princeton, NJ.
  • Dumas, B., Fleming, J. and Whaley, R. E. (1998). Implied volatility functions: Empirical tests. J. Finance 53 2059–2106.
  • Fan, J. (2005). A selective overview of nonparametric methods in financial econometrics. Statist. Sci. 20 317–357.
  • Fan, J. and Wang, Y. (2007). Multi-scale jump and volatility analysis for high-frequency financial data. J. Amer. Statist. Assoc. 102 1349–1362.
  • Fan, J. and Zhang, C. (2003). A reexamination of diffusion estimators with applications to financial model validation. J. Amer. Statist. Assoc. 98 118–134.
  • Feller, W. (1952). The parabolic differential equations and the associated semi-groups of transformations. Ann. of Math. (2) 55 468–519.
  • Friedman, A. (1964). Partial Differential Equations of Parabolic Type. Prentice Hall, Englewood Cliffs, NJ.
  • Gobet, E. (2002). LAN property for ergodic diffusions with discrete observations. Ann. Inst. Henri Poincaré Probab. Stat. 38 711–737.
  • Jacobsen, M. (2001). Discretely observed diffusions: Classes of estimating functions and small Δ-optimality. Scand. J. Statist. 28 123–149.
  • Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus. Springer, New York.
  • Merton, R. C. (1973). Theory of rational option pricing. Bell J. Econom. and Management Sci. 4 141–183.
  • Mykland, P. A. and Zhang, L. (2009). Inference for continuous semimartingales observed at high frequency. Econometrica 77 1403–1445.
  • Newey, W. K. (1991). Uniform convergence in probability and stochastic equicontinuity. Econometrica 59 1161–1167.
  • Øksendal, B. (2000). Stochastic Differential Equations: An Introduction with Applications, 5th ed. Springer, Berlin.
  • Prakasa Rao, B. L. S. (1999). Semimartingales and Statistical Inference. Chapman and Hall/CRC, London.
  • Protter, P. E. (2004). Stochastic Integration and Differential Equations: Stochastic Modelling and Applied Probability, 2nd ed. Applications of Mathematics (New York) 21. Springer, Berlin.
  • Sørensen, M. (2007). Efficient estimation for ergodic diffusions sampled at high frequency. Dept. Mathematical Sciences, Univ. Copenhagen.
  • Stramer, O. and Yan, J. (2007a). On simulated likelihood of discretely observed diffusion processes and comparison to closed-form approximation. J. Comput. Graph. Statist. 16 672–691.
  • Stramer, O. and Yan, J. (2007b). Asymptotics of an efficient Monte Carlo estimation for the transition density of diffusion processes. Methodol. Comput. Appl. Probab. 9 483–496.
  • Sundaresan, S. M. (2000). Continuous time finance: A review and assessment. J. Finance 55 1569–1622.
  • Tang, C. Y. and Chen, S. X. (2009). Parameter estimation and bias correction for diffusion processes. J. Econometrics 149 65–81.
  • Vasicek, O. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics 5 177–188.
  • Wang, Y. (2002). Asymptotic nonequivalence of Garch models and diffusions. Ann. Statist. 30 754–783.