×

Parameter estimation for fractional Ornstein-Uhlenbeck processes at discrete observation. (English) Zbl 1225.62116

Summary: This paper deals with the problem of estimating the parameters for fractional Ornstein-Uhlenbeck processes from discrete observations when the Hurst parameter \(H\) is known. Both the drift and the diffusion coefficient estimators of discrete form are obtained based on approximating integrals via Riemann sums with Hurst parameter \(H \in (1/2, 3/4)\). By adapting the stochastic integral representation to the fractional Brownian motion, these two estimators can be efficiently computed. Numerical examples are presented to examine the performance of our method. An application to real data is also presented to show how to apply this method in practice.

MSC:

62M05 Markov processes: estimation; hidden Markov models
60G22 Fractional processes, including fractional Brownian motion
60J60 Diffusion processes
65C50 Other computational problems in probability (MSC2010)

Software:

longmemo
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Prakasa Rao, B.L.S., Statistical inference for diffusion type processes, (1999), Oxford Univ. Press London · Zbl 0952.62077
[2] Duffee, G.R., Term premia and interest rate forecasts in affine models, J. finance, 57, 405-443, (2002)
[3] Dai, Q.; Singleton, K.J., Specification analysis of affine term structure models, J. finance, 55, 1943-1978, (2000)
[4] Beran, R., Statistics for long-memory processes, (1994), Chapman & Hall New York · Zbl 0869.60045
[5] Taylor, M.A.P., Maximum likelihood estimation for a road traffic network model, Appl. math. model, 5, 34-38, (1981) · Zbl 0451.90056
[6] Srivastava, P.W.; Mittal, N., Optimum step-stress partially accelerated life tests for the truncated logistic distribution with censoring, Appl. math. model, 34, 3166-3178, (2010) · Zbl 1201.90066
[7] Decreusefond, L.; Ustunel, A.S., Stochastic analysis of the fractional Brownian motion, Potential anal., 10, 177-214, (1999) · Zbl 0924.60034
[8] Norros, I.; Valkeila, E.; Virtamo, J., An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motion, Bernoulli, 5, 571-587, (1999) · Zbl 0955.60034
[9] Kleptsyna, M.L.; Le Breton, A., Statistical analysis of the fractional ornstein – uhlenbeck type process, Stat. inference stoch. process., 5, 229-248, (2002) · Zbl 1021.62061
[10] Kleptsyna, M.L.; Le Breton, A.; Roubaud, M.C., Parameter estimation and optimal filtering for fractional type stochastic systems, Stat. inference stoch. process., 3, 173-182, (2000) · Zbl 0966.62069
[11] Cialenco, I.; Lototsky, S.; Pospisil, J., Asymptotic properties of the maximum likelihood estimator for stochastic parabolic equations with additive fractional Brownian motion, Stochastics dyn., 9, 169-185, (2009) · Zbl 1176.62019
[12] Hu, Y.; Nualart, D., Parameter estimation for fractional ornstein – uhlenbeck processes, Stat. probab. lett., 80, 1030-1038, (2010) · Zbl 1187.62137
[13] Bercu, B.; Coutin, L.; Savy, N., Sharp large deviations for the fractional ornstein – uhlenbeck process, Teor. veroyatnost. i primenen., 55, 732-771, (2010) · Zbl 1252.60026
[14] Brouste, A.; Kleptsyna, M., Asymptotic properties of MLE for partially observed fractional diffusion system, Stat. inference stoch. process., 13, 1-13, (2010) · Zbl 1205.60142
[15] Brouste, A., Asymptotic properties of MLE for partially observed fractional diffusion system with dependent noises, J. stat. plann. infer., 140, 551-558, (2010) · Zbl 1177.62027
[16] Ait Sahalia, Y., Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach, Econometrica, 70, 223-262, (2002) · Zbl 1104.62323
[17] Bibby, B.; Soensen, M., Martingale estimation functions for discretely observed diffusion processes, Bernoulli, 1, 17-39, (1995) · Zbl 0830.62075
[18] Gourieroux, C.; monfort, A.; Renault, E., Indirect inference, J. appl. econometrics., 8, 85-118, (1993)
[19] Elerian, O.; Chib, S.; Shephard, N., Likelihood inference for discretely observed non-linear diffusions, Econometrica, 69, 959-993, (2001) · Zbl 1017.62068
[20] Soensen, H., Parametric inference for diffusion processes observed at discrete points in time: a survey, Int. stat. rev., 72, 337-354, (2004)
[21] Liptser, R.S.; Shiryayev, A.N., Statistics of random processes: II applications, () · Zbl 0369.60001
[22] Kutoyants, Y.A., Statistical inference for ergodic diffusion processes, (2004), Springer-Verlag London · Zbl 1038.62073
[23] Nualart, D., The Malliavin calculus and related topics, (2006), Springer-Verlag Berlin · Zbl 1099.60003
[24] Durham, G.B.; Gallant, A.R., Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes, J. bus. econ. statist., 20, 279-316, (2002)
[25] Cheridito, P.; Kawaguchi, H.; Maejima, M., Fractional ornstein – uhlenbeck processes, Electron. J. probab., 8, 1-14, (2003) · Zbl 1065.60033
[26] Mandelbrot, B.B., A statistical methodology for non-periodic cycles: from the covariance to R/S analysis, Ann. econ. soc. meas., 1, 259-290, (1972)
[27] Lo, A., Long-term memory in stock market prices, Econometrica, 59, 1279-1313, (1991) · Zbl 0781.90023
[28] Ortigueira, M.D.; Batista, A.G., A fractional linear system view of the fractional Brownian motion, Nonlinear dyn., 38, 295-303, (2004) · Zbl 1115.60044
[29] Mielniczuk, J.; Wojdyłło, P., Estimation of Hurst exponent revisited, Comput. stat. data anal., 51, 4510-4525, (2007) · Zbl 1162.62404
[30] Li, W.; Yu, C.; Carriquiry, A.; Kliemann, W., The asymptotic behavior of the R/S statistic for fractional Brownian motion, Stat. probab. lett., 81, 83-91, (2011) · Zbl 1206.62152
[31] Cheung, Y.W., Test for fractional integration: a Monte Carlo investigation, J. time ser. anal., 14, 331-345, (1993) · Zbl 0800.62546
[32] Paxson, V., Fast approximate synthesis of fractional Gaussian noise for generating self-similar network traffic, Comput. commun. rev., 27, 5-18, (1997)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.