Monte Carlo maximum likelihood estimation for discretely observed diffusion processes
Alexandros Beskos, Omiros Papaspiliopoulos, and Gareth Roberts
Source: Ann. Statist. Volume 37, Number 1
(2009), 223-245.
Abstract
This paper introduces a Monte Carlo method for maximum likelihood inference in the context of discretely observed diffusion processes. The method gives unbiased and a.s. continuous estimators of the likelihood function for a family of diffusion models and its performance in numerical examples is computationally efficient. It uses a recently developed technique for the exact simulation of diffusions, and involves no discretization error. We show that, under regularity conditions, the Monte Carlo MLE converges a.s. to the true MLE. For datasize n→∞, we show that the number of Monte Carlo iterations should be tuned as and we demonstrate the consistency properties of the Monte Carlo MLE as an estimator of the true parameter value.
First Page:
Show
Hide
Keywords: Coupling; uniform convergence; exact simulation; linear diffusion processes; random function; SLLN on Banach space
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aos/1232115933
Digital Object Identifier: doi:10.1214/07-AOS550
Mathematical Reviews number (MathSciNet): MR2488350
Zentralblatt MATH identifier: 05518693
References
[1] Aït-Sahalia, Y. (2002). Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach. Econometrica 70 223–262.
[2] Andrews, D. W. K. (1987). Consistency in nonlinear econometric models: A generic uniform law of large numbers. Econometrica 55 1465–1471.
Mathematical Reviews (MathSciNet): MR923471
Digital Object Identifier: doi:10.2307/1913568
JSTOR: links.jstor.org
[3] Asmussen, S., Glynn, P. and Pitman, J. (1995). Discretization error in simulation of one-dimensional reflecting Brownian motion. Ann. Appl. Probab. 5 875–896.
Mathematical Reviews (MathSciNet): MR1384357
Zentralblatt MATH: 0853.65147
Digital Object Identifier: doi:10.1214/aoap/1177004597
Project Euclid: euclid.aoap/1177004597
[4] Attouch, H. (1984). Variational Convergence for Functions and Operators. Pitman (Advanced Publishing Program), Boston, MA.
Mathematical Reviews (MathSciNet): MR773850
Zentralblatt MATH: 0561.49012
[5] Beck, A. (1963). On the strong law of large numbers. In Ergodic Theory (Proc. Internat. Sympos., Tulane Univ., New Orleans, La., 1961) 21–53. Academic Press, New York.
Mathematical Reviews (MathSciNet): MR160256
Zentralblatt MATH: 0132.13001
[6] Beskos, A., Papaspiliopoulos, O. and Roberts, G. O. (2008). A factorisation of diffusion measure and finite sample path constructions. Methodol. Comput. Appl. Probab. 10 85–104.
Mathematical Reviews (MathSciNet): MR2394037
Zentralblatt MATH: 1152.65013
Digital Object Identifier: doi:10.1007/s11009-007-9060-4
[7] Beskos, A., Papaspiliopoulos, O. and Roberts, G. O. (2006). Retrospective exact simulation of diffusion sample paths with applications. Bernoulli 12 1077–1098.
Mathematical Reviews (MathSciNet): MR2274855
Digital Object Identifier: doi:10.3150/bj/1165269151
Project Euclid: euclid.bj/1165269151
[8] 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 1–29.
Mathematical Reviews (MathSciNet): MR2278331
Zentralblatt MATH: 1100.62079
Digital Object Identifier: doi:10.1111/j.1467-9868.2006.00552.x
[9] Beskos, A. and Roberts, G. O. (2005). Exact simulation of diffusions. Ann. Appl. Probab. 15 2422–2444.
Mathematical Reviews (MathSciNet): MR2187299
Zentralblatt MATH: 1101.60060
Digital Object Identifier: doi:10.1214/105051605000000485
Project Euclid: euclid.aoap/1133965767
[10] Bibby, B. M. and Sørensen, M. (1995). Martingale estimation functions for discretely observed diffusion processes. Bernoulli 1 17–39.
[11] Burke, G. (1965). A uniform ergodic theorem. Ann. Math. Statist. 36 1853–1858.
Mathematical Reviews (MathSciNet): MR183001
Zentralblatt MATH: 0249.60012
Digital Object Identifier: doi:10.1214/aoms/1177699815
Project Euclid: euclid.aoms/1177699815
[12] Cappé, O., Douc, R., Moulines, E. and Robert, C. (2002). On the convergence of the Monte Carlo maximum likelihood method for latent variable models. Scand. J. Statist. 29 615–635.
Mathematical Reviews (MathSciNet): MR1988415
Digital Object Identifier: doi:10.1111/1467-9469.00309
[13] Choirat, C., Hess, C. and Seri, R. (2003). A functional version of the Birkhoff ergodic theorem for a normal integrand: A variational approach. Ann. Probab. 31 63–92.
Mathematical Reviews (MathSciNet): MR1959786
Zentralblatt MATH: 1015.60029
Digital Object Identifier: doi:10.1214/aop/1046294304
Project Euclid: euclid.aop/1046294304
[14] Devroye, L. (1986). Nonuniform Random Variate Generation. Springer, New York.
Mathematical Reviews (MathSciNet): MR836973
[15] Dudley, R. M. (1999). Uniform Central Limit Theorems. Cambridge Univ. Press.
Mathematical Reviews (MathSciNet): MR1720712
[16] Durham, G. B. and Gallant, A. R. (2002). Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes (with discussion). J. Bus. Econom. Statist. 20 297–338.
Mathematical Reviews (MathSciNet): MR1939904
JSTOR: links.jstor.org
[17] 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
[18] 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
[19] Ferguson, T. S. (1996). A Course in Large Sample Theory. Chapman and Hall, London.
Mathematical Reviews (MathSciNet): MR1699953
Zentralblatt MATH: 0871.62002
[20] Gallant, A. R. and Tauchen, G. (1996). Which moments to match? Econometric Theory 12 657–681.
Mathematical Reviews (MathSciNet): MR1422547
Digital Object Identifier: doi:10.1017/S0266466600006976
JSTOR: links.jstor.org
[21] Genon-Catalot, V. and Jacod, J. (1993). On the estimation of the diffusion coefficient for multi-dimensional diffusion processes. Ann. Inst. H. Poincaré Probab. Statist. 29 119–151.
Mathematical Reviews (MathSciNet): MR1204521
Zentralblatt MATH: 0770.62070
[22] Geyer, C. J. (1994). On the convergence of Monte Carlo maximum likelihood calculations. J. Roy. Statist. Soc. Ser. B 56 261–274.
Mathematical Reviews (MathSciNet): MR1257812
JSTOR: links.jstor.org
[23] Geyer, C. J. and Thompson, E. A. (1992). Constrained Monte Carlo maximum likelihood for dependent data (with discussion). J. Roy. Statist. Soc. Ser. B 54 657–699.
Mathematical Reviews (MathSciNet): MR1185217
JSTOR: links.jstor.org
[24] Giesy, D. P. (1976). Strong laws of large numbers for independent sequences of Banach space-valued random variables. In Probability in Banach Spaces (Proc. First Internat. Conf., Oberwolfach, 1975) 89–99. Lecture Notes in Math. 526. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR440673
Zentralblatt MATH: 0366.60034
[25] Gourieroux, C., Monfort, A. and Renault, E. (1993). Indirect inference. J. Applied Econometrics 8 85–118.
[26] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.
Mathematical Reviews (MathSciNet): MR624435
Zentralblatt MATH: 0462.60045
[27] Kessler, M. and Paredes, S. (2002). Computational aspects related to martingale estimating functions for a discretely observed diffusion. Scand. J. Statist. 29 425–440.
Mathematical Reviews (MathSciNet): MR1925568
Digital Object Identifier: doi:10.1111/1467-9469.00299
[28] Kingman, J. F. C. (1993). Poisson Processes. Oxford Univ. Press, New York.
Mathematical Reviews (MathSciNet): MR1207584
[29] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR1214374
[30] Mourier, E. (1953). Eléments aléatoires dans un espace de Banach. Ann. Inst. H. Poincaré 13 161–244.
Mathematical Reviews (MathSciNet): MR64339
[31] Øksendal, B. (2003). Stochastic Differential Equations, 6th ed. Springer, Berlin.
[32] Pedersen, A. R. (1995). Consistency and asymptotic normality of an approximate maximum likelihood estimator for discretely observed diffusion processes. Bernoulli 1 257–279.
Mathematical Reviews (MathSciNet): MR1363541
Digital Object Identifier: doi:10.2307/3318480
Project Euclid: euclid.bj/1193667818
[33] Peskir, G. (2000). From Uniform Laws of Large Numbers to Uniform Ergodic Theorems. Univ. Aarhus, Dept. Mathematics.
Mathematical Reviews (MathSciNet): MR1805157
Zentralblatt MATH: 0992.37004
[34] Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (1992). Numerical Recipes in C, 2nd ed. Cambridge Univ. Press.
Mathematical Reviews (MathSciNet): MR1201159
[35] Roberts, G. O. and Stramer, O. (2001). On inference for partially observed nonlinear diffusion models using the Metropolis–Hastings algorithm. Biometrika 88 603–621.
[36] Shepp, L. A. (1979). The joint density of the maximum and its location for a Wiener process with drift. J. Appl. Probab. 16 423–427.
Mathematical Reviews (MathSciNet): MR531776
Zentralblatt MATH: 0403.60072
Digital Object Identifier: doi:10.2307/3212910
JSTOR: links.jstor.org
[37] Sørensen, H. (2004). Parametric inference for diffusion processes observed at discrete points in time: A survey. Internat. Statist. Rev. 72 337–354.
[38] Talagrand, M. (1987). The Glivenko–Cantelli problem. Ann. Probab. 15 837–870.
Mathematical Reviews (MathSciNet): MR893902
Zentralblatt MATH: 0632.60024
Digital Object Identifier: doi:10.1214/aop/1176992069
Project Euclid: euclid.aop/1176992069
[39] Tucker, H. G. (1959). A generalization of the Glivenko–Cantelli theorem. Ann. Math. Statist. 30 828–830.
Mathematical Reviews (MathSciNet): MR107891
Digital Object Identifier: doi:10.1214/aoms/1177706212
Project Euclid: euclid.aoms/1177706212
[40] Wald, A. (1949). Note on the consistency of the maximum likelihood estimate. Ann. Math. Statistics 20 595–601.
Mathematical Reviews (MathSciNet): MR32169
Zentralblatt MATH: 0034.22902
Digital Object Identifier: doi:10.1214/aoms/1177729952
Project Euclid: euclid.aoms/1177729952
