Globally optimal parameter estimates for nonlinear diffusions
Aleksandar Mijatović and Paul Schneider
Source: Ann. Statist. Volume 38, Number 1
(2010), 215-245.
Abstract
This paper studies an approximation method for the log-likelihood function of a nonlinear diffusion process using the bridge of the diffusion. The main result (Theorem 1) shows that this approximation converges uniformly to the unknown likelihood function and can therefore be used efficiently with any algorithm for sampling from the law of the bridge. We also introduce an expected maximum likelihood (EML) algorithm for inferring the parameters of discretely observed diffusion processes. The approach is applicable to a subclass of nonlinear SDEs with constant volatility and drift that is linear in the model parameters. In this setting, globally optimal parameters are obtained in a single step by solving a linear system. Simulation studies to test the EML algorithm show that it performs well when compared with algorithms based on the exact maximum likelihood as well as closed-form likelihood expansions.
First Page:
Show
Hide
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aos/1262271614
Digital Object Identifier: doi:10.1214/09-AOS710
Mathematical Reviews number (MathSciNet): MR2589321
Zentralblatt MATH identifier: 1181.62121
References
Aït-Sahalia, Y. (1996). Testing continuous-time models of the spot interest rate. Review of 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. (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.
Andersen, T. G., Benzoni, L. and Lund, J. (2002). An empirical investigation of continuous-time equity return models. J. Finance 57 1239–1284.
Beskos, A., Papaspiliopoulos, O., Roberts, G. and Fearnhead, P. (2006). Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 333–382.
Mathematical Reviews (MathSciNet): MR2278331
Zentralblatt MATH: 1100.62079
Digital Object Identifier: doi:10.1111/j.1467-9868.2006.00552.x
Bladt, M. and Sørensen, M. (2007). Simple simulation of diffusion bridges with application to likelihood inference for diffusions. Working paper, CAF Centre for Analytical Finance, Univ. Aarhus.
Brandt, M. W. and Santa-Clara, P. (2002). Simulated likelihood estimation of diffusions with an application to exchange rate dynamics in incomplete markets. Journal of Financial Economics 63 161–210.
Chib, S. and Shephard, N. (2002). Comment on Garland B. Durham and A. Ronald Gallant’s “numerical techniques for maximum likelihood estimation of continuous-time diffusion processes.” J. Bus. Econom. Statist. 20 325–327.
Mathematical Reviews (MathSciNet): MR1939904
JSTOR: links.jstor.org
Dempster, A., Laird, N. and Rubin, D. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). J. Roy. Statist. Soc. Ser. B 39 1–38.
Mathematical Reviews (MathSciNet): MR501537
JSTOR: links.jstor.org
Durham, G. B. and Gallant, R. A. (2002). Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes. J. Bus. Econom. Statist. 20 297–316.
Mathematical Reviews (MathSciNet): MR1939904
JSTOR: links.jstor.org
Elerian, O., Chib, S. and Shephard, N. (2001). Likelihood inference for discretely observed nonlinear diffusions. Econometrica 69 959–993.
Mathematical Reviews (MathSciNet): MR1839375
Digital Object Identifier: doi:10.1111/1468-0262.00226
JSTOR: links.jstor.org
Eraker, B. (2001). MCMC analysis of diffusion models with application to finance. J. Bus. Econom. Statist. 19 177–191.
Mathematical Reviews (MathSciNet): MR1939708
JSTOR: links.jstor.org
Gallant, A. R. and Tauchen, G. (2006). EMM: A program for efficient method of moments estimation. Duke Univ. Available at http://econ.duke.edu/webfiles/arg/emm.
Gallant, A. R. and Tauchen, G. (2009). Simulated score methods and indirect inference for continuous-time models. Elsevier.
Hurn, A., Jeisman, J. and Lindsay, K. (2007). Seeing the wood for the trees: A critical evaluation of methods to estimate the parameters of stochastic differential equations. Journal of Financial Econometrics 5 390–455.
Jensen, B. and Poulsen, R. (2002). Transition densities of diffusion processes: Numerical comparison of approximation techniques. Journal of Derivatives 9 18–32.
Jones, C. S. (1998). Bayesian estimation of continuous-time finance models. Working paper, Univ. Rochester.
Jones, C. S. (2003a). The dynamics of stochastic volatility: Evidence from underlying and options markets. J. Econometrics 116 181–224.
Mathematical Reviews (MathSciNet): MR2002525
Zentralblatt MATH: 1016.62122
Digital Object Identifier: doi:10.1016/S0304-4076(03)00107-6
Jones, C. S. (2003b). Nonlinear mean reversion in the short-term interest rate. Review of Financial Studies 16 793–843.
Kloeden, P. E. and Platen, E. (1999). Numerical Solution of Stochastic Differential Equations. Springer, New York.
McLachlan, G. J. and Krishnan, T. (1997). The EM Algorithm and Extensions. Wiley, New York.
Mathematical Reviews (MathSciNet): MR1417721
Papaspiliopoulos, O. and Sermaidis, G. (2007). Monotonicity properties of the Monte Carlo EM algorithm and connections with simulated likelihood. Working Paper, Warwick Univ.
Pedersen, A. (1995). A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scand. J. Statist. 22 55–71.
Mathematical Reviews (MathSciNet): MR1334067
Roberts, G. O. and Stramer, O. (2001). On inference for partially observed nonlinear diffusion models using the metropolis-hastings algorithm. Biometrika 88 603–621.
Mathematical Reviews (MathSciNet): MR1859397
Zentralblatt MATH: 0985.62066
Digital Object Identifier: doi:10.1093/biomet/88.3.603
JSTOR: links.jstor.org
Rogers, L. (1985). Smooth transition densities for one-dimensional diffusions. Bull. Lond. Math. Soc. 17 157–161.
Mathematical Reviews (MathSciNet): MR806242
Zentralblatt MATH: 0604.60077
Digital Object Identifier: doi:10.1112/blms/17.2.157
Schneider, P. (2006). Approximations of transition densities for nonlinear multivariate diffusions with an application to dynamic term structure models. Working paper, Vienna Univ. Economics and Business Administration.
Sørensen, H. (2004). Parametric inference for diffusion processes observed at discrete points: A survey. International Statistical Review 72 337–354.
Stramer, O. and Yan, J. (2007a). Asymptotics of an efficient Monte Carlo estimation for the transition density of diffusion processes. Methodol. Comput. Appl. Probab. 9 483–496.
Mathematical Reviews (MathSciNet): MR2404739
Zentralblatt MATH: 1134.60369
Digital Object Identifier: doi:10.1007/s11009-006-9006-2
Stramer, O. and Yan, J. (2007b). On simulated likelihood of discretely observed diffusion processes and comparison to closed-form approximation. J. Comput. Graph. Statist. 16 672–691.
Mathematical Reviews (MathSciNet): MR2351085
Digital Object Identifier: doi:10.1198/106186007X237306
