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Statistical aspects of the fractional stochastic calculus. (English) Zbl 1194.62097

Summary: We apply the techniques of stochastic integration with respect to fractional Brownian motion and the theory of regularity and supremum estimation for stochastic processes to study the maximum likelihood estimator (MLE) for the drift parameter of stochastic processes satisfying stochastic equations driven by a fractional Brownian motion with any level of Hölder-regularity (any Hurst parameter). We prove existence and strong consistency of the MLE for linear and nonlinear equations. We also prove that a version of the MLE using only discrete observations is still a strongly consistent estimator.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62M09 Non-Markovian processes: estimation
62H05 Characterization and structure theory for multivariate probability distributions; copulas
60H30 Applications of stochastic analysis (to PDEs, etc.)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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[1] Aït Sahalia, Y. (2002). Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach. Econometrica 70 223–262. JSTOR: · Zbl 1104.62323 · doi:10.1111/1468-0262.00274
[2] Alòs, E., Mazet, O. and Nualart, D. (2001). Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29 766–801. · Zbl 1015.60047 · doi:10.1214/aop/1008956692
[3] Bardet, J.-M., Lang, G., Oppenheim, G., Philippe, A., Stoev, S. and Taqqu, M. S. (2003). Semi-parametric estimation of the long-range dependence parameter: A survey. In Theory and Applications of Long-Range Dependence (P. Doukhan, G. Oppenheim and M. S. Taqqu, eds.) 557–577. Birkhäuser, Boston. · Zbl 1032.62077
[4] Basawa, I. V. and Prakasa Rao, B. L. S. (1980). Statistical Inference for Stochastic Processes. Academic Press, London. · Zbl 0448.62070
[5] Beskos, A., Papaspiliopoulos, O., Roberts, G. 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 · doi:10.1111/j.1467-9868.2006.00552.x
[6] Bibby, B. and Sørensen, M. (1995). Martingale estimation functions for discretely observed diffusion processes. Bernoulli 1 17–39. · Zbl 0830.62075 · doi:10.2307/3318679
[7] Bishwal, J. P. and Bose, A. (2001). Rates of convergence of approximate maximum likelihood estimators in the Ornstein–Uhlenbeck process. Comput. Math. Appl. 42 23–38. · Zbl 0979.62061 · doi:10.1016/S0898-1221(01)00127-4
[8] Boufoussi, B. and Ouknine, Y. (2003). On a SDE driven by a fractional Brownian motion and with monotone drift. Electron. Comm. Probab. 8 122–134. · Zbl 1060.60060
[9] Cheridito, P., Kawaguchi, H. and Maejima, M. (2003). Fractional Ornstein–Uhlenbeck processes. Electron. J. Probab. 8 14 pp. · Zbl 1065.60033
[10] Cleur, E. M. (2001). Maximum likelihood estimates of a class of one-dimensional stochastic differential equation models from discrete data. J. Time Ser. Anal. 22 505–515. · Zbl 0979.62062 · doi:10.1111/1467-9892.00238
[11] Coeurjolly, J.-F. (2005). L-type estimators of the fractal dimension of locally self-similar Gaussian processes. Preprint. Available at hal.ccsd.cnrs.fr/docs/00/03/13/77/PDF/robustHurstHAL.pdf.
[12] 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 · doi:10.1080/17442508608833428
[13] Decreusefond, L. and Üstünel, A. S. (1999). Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 177–214. · Zbl 0924.60034 · doi:10.1023/A:1008634027843
[14] Djehiche, B. and Eddahbi, M. (2001). Hedging options in market models modulated by fractional Brownian motion. Stochastic Anal. Appl. 19 753–770. · Zbl 0993.60040 · doi:10.1081/SAP-120000220
[15] Elerian, O., Chib, S. and Shephard, N. (2001). Likelihood inference for discretely observed nonlinear diffusions. Econometrica 69 959–993. JSTOR: · Zbl 1017.62068 · doi:10.1111/1468-0262.00226
[16] Gourieroux, C., Monfort, A. and Renault, E. (1993). Indirect inference. J. Appl. Econometrics 8 85–118.
[17] Hu, Y. and Øksendhal, B. (2003). Fractional white noise calculus and applications to finance. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 1–32. · Zbl 1045.60072 · doi:10.1142/S0219025703001110
[18] Kleptsyna, M. L. and Le Breton, A. (2002). Statistical analysis of the fractional Ornstein–Uhlenbeck type process. Statist. Inference Stoch. Process . 5 229–248. · Zbl 1021.62061 · doi:10.1023/A:1021220818545
[19] Kleptsyna, M., Le Breton, A. and Roubaud, M.-C. (2000). Parameter estimation and optimal filtering for fractional type stochastic systems. Statist. Inference Stoch. Process . 3 173–182. · Zbl 0966.62069 · doi:10.1023/A:1009923431187
[20] Kukush, A., Mishura, Y. and Valkeila, E. (2005). Statistical inference with fractional Brownian motion. Statist. Inference Stoch. Process . 8 71–93. · Zbl 1107.62355 · doi:10.1023/B:SISP.0000049124.59173.79
[21] Kutoyants, Yu. A. (1977). On a property of estimator of parameter of trend coefficient. Izv. Akad. Nauk Arm. SSR. Mathematika 12 245–251. · Zbl 0405.60066
[22] Kutoyants, Yu. A. (1977). Estimation of the trend parameter of a diffusion process in the smooth case. Theory Probab. Appl. 22 399–405. · Zbl 0379.62072 · doi:10.1137/1122047
[23] Kutoyants, Yu. A. (1980). Parameter Estimation for Stochastic Processes . Heldermann, Berlin. · Zbl 0542.62073
[24] Kutoyants, Yu. A. (2004). Statistical Inference for Ergodic Diffusion Processes. Springer, London. · Zbl 1038.62073
[25] Le Breton, A. (1976). On continuous and discrete sampling for parameter estimation in diffusion type processes. In Stochastic Systems : Modeling , Identification and Optimization 1 (R. J.-B. Wets, ed.) 124–144. North-Holland, Amsterdam. · Zbl 0368.93034
[26] Liptser, R. S. and Shiryayev, A. N. (1978). Statistics of Random Processes 2 . Applications . Springer, Berlin. · Zbl 0369.60001
[27] Malliavin, P. (1976). Stochastic calculus of variation and hypoelliptic operators. In Proc. Internat. Symp. on Stochastic Differential Equations (K. Itô, ed.) 195–263. Wiley, New York. · Zbl 0411.60060
[28] Norros, I., Valkeila, E. and Virtamo, J. (1999). An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5 571–587. · Zbl 0955.60034 · doi:10.2307/3318691
[29] Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, New York. · Zbl 0837.60050
[30] Nualart, D. and Ouknine, Y. (2002). Regularization of differential equations by fractional noise. Stochastic Process. Appl. 102 103–116. · Zbl 1075.60536 · doi:10.1016/S0304-4149(02)00155-2
[31] Pedersen, A. R. (1995). A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scand. J. Statist. 22 55–71. · Zbl 0827.62087
[32] Poulsen, R. (1999). Approximate maximum likelihood estimation of discretely observed diffusion processes. Working Paper 29, Centre for Analytical Finance, Univ. Aarhus.
[33] Prakasa Rao, B. L. S. (2003). Parametric estimation for linear stochastic differential equations driven by fractional Brownian motion. Random Oper. Stochastic Equations 11 229–242. · Zbl 1053.62089 · doi:10.1163/156939703771378581
[34] Prakasa Rao, B. L. S. (1999). Statistical Inference for Diffusion Type Processes. Oxford Univ. Press. · Zbl 0952.62077
[35] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion , 3rd ed. Springer, Berlin. · Zbl 0917.60006
[36] Sørensen, H. (2004). Parametric inference for diffusion processes observed at discrete points in time: A survey. Internat. Statist. Rev. 72 337–354.
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