×

Maximum-likelihood estimation for diffusion processes via closed-form density expansions. (English) Zbl 1273.62196

Summary: This paper proposes a widely applicable method of approximate maximum-likelihood estimation for multivariate diffusion process from discretely sampled data. A closed-form asymptotic expansion for the transition density is proposed and accompanied by an algorithm containing only basic and explicit calculations for delivering any arbitrary order of the expansion. The likelihood function is thus approximated explicitly and employed in statistical estimation.
The performance of our method is demonstrated by Monte Carlo simulations from implementing several examples, which represent a wide range of commonly used diffusion models. The convergence related to the expansion and the estimation method are theoretically justified using the theory of S. Watanabe [Ann. Probab. 15, 1–39 (1987; Zbl 0633.60077)] and N. Yoshida [J. Jap. Stat. Soc. 22, No. 2, 139–159 (1992; Zbl 0778.62018)] on the analysis of the generalized random variables under some standard sufficient conditions.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62F12 Asymptotic properties of parametric estimators
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
62H12 Estimation in multivariate analysis
65C05 Monte Carlo methods
60H30 Applications of stochastic analysis (to PDEs, etc.)
65C60 Computational problems in statistics (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] Aït-Sahalia, Y. (1999). Transition densities for interest rate and other nonlinear diffusions. J. Finance 54 1361-1395.
[2] Aït-Sahalia, Y. (2002). Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach. Econometrica 70 223-262. · Zbl 1104.62323
[3] Aït-Sahalia, Y. (2008). Closed-form likelihood expansions for multivariate diffusions. Ann. Statist. 36 906-937. · Zbl 1246.62180
[4] Aït-Sahalia, Y. (2009). Estimating and testing continuous-time models in finance: The role of transition densities. Annual Review of Financial Economics 1 341-359.
[5] Aït-Sahalia, Y. and Kimmel, R. (2007). Maximum likelihood estimation of stochastic volatility models. Journal of Financial Economics 83 413-452.
[6] 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.
[7] Aït-Sahalia, Y. and Mykland, P. A. (2003). The effects of random and discrete sampling when estimating continuous-time diffusions. Econometrica 71 483-549. · Zbl 1142.60381
[8] Aït-Sahalia, Y. and Mykland, P. A. (2004). Estimators of diffusions with randomly spaced discrete observations: A general theory. Ann. Statist. 32 2186-2222. · Zbl 1062.62155
[9] Aït-Sahalia, Y. and Yu, J. (2006). Saddlepoint approximations for continuous-time Markov processes. J. Econometrics 134 507-551. · Zbl 1418.62286
[10] Azencott, R. (1984). Densité des diffusions en temps petit: Développements asymptotiques. In Lecture Notes in Math. Seminar on Probability XVIII 1059 402-498. Springer, Berlin. · Zbl 0546.60079
[11] Bakshi, G. and Ju, N. (2005). A refinement to Aït-Sahalia’s (2003) “Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach.” J. Bus. 78 2037-2052.
[12] Bakshi, G., Ju, N. and Ou-Yang, H. (2006). Estimation of continuous-time models with an application to equity volatility dynamics. Journal of Financial Economics 82 227-249.
[13] Basawa, I. V. and Prakasa Rao, B. L. S. (1980). Statistical Inference for Stochastic Processes : Probability and Mathematical Statistics . Academic Press, London. · Zbl 0448.62070
[14] Ben Arous, G. (1988). Methods de Laplace et de la phase stationnaire sur l’espace de Wiener. Stochastics 25 125-153. · Zbl 0666.60026
[15] Beskos, A., Papaspiliopoulos, O. and Roberts, G. (2009). Monte Carlo maximum likelihood estimation for discretely observed diffusion processes. Ann. Statist. 37 223-245. · Zbl 1169.65004
[16] 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. · Zbl 1100.62079
[17] Beskos, A. and Roberts, G. O. (2005). Exact simulation of diffusions. Ann. Appl. Probab. 15 2422-2444. · Zbl 1101.60060
[18] Bhattacharya, R. N. and Ghosh, J. K. (1978). On the validity of the formal Edgeworth expansion. Ann. Statist. 6 434-451. · Zbl 0396.62010
[19] Bibby, B. M. and Sørensen, M. (1995). Martingale estimation functions for discretely observed diffusion processes. Bernoulli 1 17-39. · Zbl 0830.62075
[20] Bismut, J.-M. (1984). Large Deviations and the Malliavin Calculus. Progress in Mathematics 45 . Birkhäuser, Boston, MA. · Zbl 0537.35003
[21] 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.
[22] Chang, J. and Chen, S. X. (2011). On the approximate maximum likelihood estimation for diffusion processes. Ann. Statist. 39 2820-2851. · Zbl 1246.62181
[23] Dacunha-Castelle, D. and Florens-Zmirou, D. (1986). Estimation of the coefficients of a diffusion from discrete observations. Stochastics 19 263-284. · Zbl 0626.62085
[24] Dai, Q. and Singleton, K. J. (2000). Specification analysis of affine term structure models. J. Finance 55 1943-1978.
[25] Durham, G. B. and Gallant, A. R. (2002). Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes. J. Bus. Econom. Statist. 20 297-338. With comments and a reply by the authors.
[26] Egorov, A. V., Li, H. and Xu, Y. (2003). Maximum likelihood estimation of time-inhomogeneous diffusions. J. Econometrics 114 107-139. · Zbl 1085.62131
[27] Elerian, O. (1998). A note on the existence of a closed form conditional transition density for the Milstein scheme. Working papers, Univ. Oxford Economics.
[28] Elerian, O., Chib, S. and Shephard, N. (2001). Likelihood inference for discretely observed nonlinear diffusions. Econometrica 69 959-993. · Zbl 1017.62068
[29] Feller, W. (1951). Two singular diffusion problems. Ann. of Math. (2) 54 173-182. · Zbl 0045.04901
[30] Feller, W. (1971). An Introduction to Probability Theory and Its Applications , Vol. 2. Wiley, New York. · Zbl 0219.60003
[31] Filipović, D., Mayerhofer, E. and Schneider, P. (2011). Density approximations for multivariate affine jump-diffusion processes. · Zbl 1284.62110
[32] Genon-Catalot, V. and Jacod, J. (1993). On the estimation of the diffusion coefficient for multi-dimensional diffusion processes. Ann. Inst. Henri Poincaré Probab. Stat. 29 119-151. · Zbl 0770.62070
[33] Hall, P. (1995). The Bootstrap and Edgeworth Expansion . Springer, New York. · Zbl 0744.62026
[34] 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.
[35] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes , 2nd ed. North-Holland Mathematical Library 24 . North-Holland, Amsterdam. · Zbl 0684.60040
[36] Jensen, B. and Poulsen, R. (2002). Transition densities of diffusion processes: Numerical comparison of approximation techniques. Journal of Derivatives 9 1-15.
[37] Kanwal, R. P. (2004). Generalized Functions : Theory and Applications , 3rd ed. Birkhäuser, Boston, MA. · Zbl 1069.46001
[38] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus , 2nd ed. Graduate Texts in Mathematics 113 . Springer, New York. · Zbl 0734.60060
[39] Kessler, M. (1997). Estimation of an ergodic diffusion from discrete observations. Scand. J. Stat. 24 211-229. · Zbl 0879.60058
[40] Kessler, M. and Sørensen, M. (1999). Estimating equations based on eigenfunctions for a discretely observed diffusion process. Bernoulli 5 299-314. · Zbl 0980.62074
[41] Kloeden, P. E. and Platen, E. (1999). The Numerical Solution of Stochastic Differential Equations . Springer, New York. · Zbl 0752.60043
[42] Kunitomo, N. and Takahashi, A. (2001). The asymptotic expansion approach to the valuation of interest rate contingent claims. Math. Finance 11 117-151. · Zbl 0994.91023
[43] Léandre, R. (1988). Applications quantitatives et géométriques du calcul de Malliavin. In Stochastic Analysis ( Paris , 1987) (M. Metivier and S. Watanabe, eds.). Lecture Notes in Math. 1322 109-133. Springer, Berlin. · Zbl 0666.60014
[44] Li, C. (2013). Supplement to “Maximum-likelihood estimation for diffusion processes via closed-form density expansions.” . · Zbl 1273.62196
[45] Li, M. (2010). A damped diffusion framework for financial modeling and closed-form maximum likelihood estimation. J. Econom. Dynam. Control 34 132-157. · Zbl 1182.91211
[46] Lo, A. W. (1988). Maximum likelihood estimation of generalized Itô processes with discretely sampled data. Econometric Theory 4 231-247.
[47] McCullagh, P. (1987). Tensor Methods in Statistics . Chapman & Hall, London. · Zbl 0732.62003
[48] Mykland, P. A. (1992). Asymptotic expansions and bootstrapping distributions for dependent variables: A martingale approach. Ann. Statist. 20 623-654. · Zbl 0759.62011
[49] Mykland, P. A. (1993). Asymptotic expansions for martingales. Ann. Probab. 21 800-818. · Zbl 0776.60047
[50] Mykland, P. A. (1994). Bartlett type identities for martingales. Ann. Statist. 22 21-38. · Zbl 0808.62030
[51] Mykland, P. A. (1995). Martingale expansions and second order inference. Ann. Statist. 23 707-731. · Zbl 0839.62083
[52] Mykland, P. A. and Zhang, L. (2010). The econometrics of high frequency data. In Statistical Methods for Stochastic Differential Equations (M. Kessler, A. Lindner and M. Sørensen, eds.). Chapman & Hall/CRC, Boca Raton, FL. · Zbl 1375.62023
[53] Nualart, D. (2006). The Malliavin Calculus and Related Topics , 2nd ed. Springer, Berlin. · Zbl 1099.60003
[54] Nualart, D., Üstünel, A. S. and Zakai, M. (1988). On the moments of a multiple Wiener-Itô integral and the space induced by the polynomials of the integral. Stochastics 25 233-240. · Zbl 0669.60052
[55] Pedersen, A. R. (1995). A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scand. J. Stat. 22 55-71. · Zbl 0827.62087
[56] Phillips, P. C. and Yu, J. (2009). Maximum likelihood and Gaussian estimation of continuous time models in finance. In Handbook of Financial Time Series (T. Andersen, R. Davis, J. P. Kreib and T. Mikosch, eds.). Springer, New York. · Zbl 1178.91230
[57] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion , 3rd ed. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 293 . Springer, Berlin. · Zbl 0917.60006
[58] Schaumburg, E. (2001). Maximum likelihood estimation of jump processes with applications to finance. Ph.D. thesis, Princeton Univ.
[59] Sørensen, M. (2012). Estimating functions for diffusion-type processes. In Statistical Methods for Stochastic Differential Equations (M. Kessler, A. Lindner and M. Sørensen, eds.). Monographs on Statistics and Applied Probability 124 1-107. Chapman and Hall/CRC, Boca Raton, FL. · Zbl 1375.60124
[60] Stramer, O. and Yan, J. (2007). On simulated likelihood of discretely observed diffusion processes and comparison to closed-form approximation. J. Comput. Graph. Statist. 16 672-691.
[61] Tang, C. Y. and Chen, S. X. (2009). Parameter estimation and bias correction for diffusion processes. J. Econometrics 149 65-81. · Zbl 1429.62370
[62] Tocino, A. (2009). Multiple stochastic integrals with Mathematica. Math. Comput. Simulation 79 1658-1667. · Zbl 1159.65304
[63] Uchida, M. and Yoshida, N. (2012). Adaptive estimation of an ergodic diffusion process based on sampled data. Stochastic Process. Appl. 122 2885-2924. · Zbl 1243.62113
[64] Uemura, H. (1987). On a short time expansion of the fundamental solution of heat equations by the method of Wiener functionals. J. Math. Kyoto Univ. 27 417-431. · Zbl 0638.60067
[65] Watanabe, S. (1987). Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels. Ann. Probab. 15 1-39. · Zbl 0633.60077
[66] Xiu, D. (2011). Dissecting and deciphering European option prices using closed-form series expansion. Technical report, Univ. Chicago Booth School of Business.
[67] Yoshida, N. (1992). Asymptotic expansion for statistics related to small diffusions. J. Japan Statist. Soc. 22 139-159. · Zbl 0778.62018
[68] Yoshida, N. (1992). Asymptotic expansions of maximum likelihood estimators for small diffusions via the theory of Malliavin-Watanabe. Probab. Theory Related Fields 92 275-311. · Zbl 0767.60035
[69] Yoshida, N. (1992). Estimation for diffusion processes from discrete observation. J. Multivariate Anal. 41 220-242. · Zbl 0811.62083
[70] Yoshida, N. (1997). Malliavin calculus and asymptotic expansion for martingales. Probab. Theory Related Fields 109 301-342. · Zbl 0888.60020
[71] Yoshida, N. (2001). Malliavin calculus and martingale expansion. Bull. Sci. Math. 125 431-456. · Zbl 0998.60055
[72] Yu, J. (2007). Closed-form likelihood approximation and estimation of jump-diffusions with an application to the realignment risk of the Chinese Yuan. J. Econometrics 141 1245-1280. · Zbl 1418.62291
[73] Yu, J. and Phillips, P. C. (2001). A Gaussian approach for estimating continuouse time models of short term interest rates. Econom. J. 4 211-225. · Zbl 1051.91516
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.