The Library

Globally optimal parameter estimates for nonlinear diffusions

Tools

Mijatović, Aleksandar and Schneider, Paul. (2010) Globally optimal parameter estimates for nonlinear diffusions. Annals of statistics, Vol.38 (No.1). pp. 215-245. ISSN 0090-5364

Preview

PDF
WRAP_Mijatovic_Globally_optimal.pdf - Published Version - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
Download (427Kb)

Official URL: http://dx.doi.org/10.1214/09-AOS710

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 its closed-form likelihood expansions.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Social Sciences > Warwick Business School
Library of Congress Subject Headings (LCSH):	Diffusion processes
Journal or Publication Title:	Annals of statistics
Publisher:	Inst Mathematical Statistics
ISSN:	0090-5364
Date:	February 2010
Volume:	Vol.38
Number:	No.1
Number of Pages:	31
Page Range:	pp. 215-245
Identification Number:	10.1214/09-AOS710
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access
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. MR1926260 AÏT-SAHALIA, Y. (2008). Closed-form likelihood expansions for multivariate diffusions. Ann. Statist. 36 906–937. MR2396819 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 continuoustime 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. MR2278331 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. MR1939904 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. MR0501537 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. MR1939904 ELERIAN, O., CHIB, S. and SHEPHARD, N. (2001). Likelihood inference for discretely observed nonlinear diffusions. Econometrica 69 959–993. MR1839375 ERAKER, B. (2001). MCMC analysis of diffusion models with application to finance. J. Bus. Econom. Statist. 19 177–191. MR1939708 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. MR2002525 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. 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. 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. MR1859397 ROGERS, L. (1985). Smooth transition densities for one-dimensional diffusions. Bull. Lond. Math. Soc. 17 157–161. MR0806242 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. andYAN, J. (2007a). Asymptotics of an efficientMonte Carlo estimation for the transition density of diffusion processes. Methodol. Comput. Appl. Probab. 9 483–496. MR2404739 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. MR2351085
URI:	http://wrap.warwick.ac.uk/id/eprint/16565