Estimation in models driven by fractional brownian motion
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 2, pp. 191-213.

Soit bH(t),t le mouvement Brownien fractionnaire de paramètre 0<H<1. Lorsque 1/2<H, nous considérons des équations de diffusion de la forme

X(t)=c+0tσ(X(u))dbH(u)+0tμ(X(u))du.
Nous proposons dans des modèles particuliers où, σ(x)=σ ou σ(x)=σx et μ(x)=μ ou μ(x)=μx, un théorème central limite pour des estimateurs de H et de σ, obtenus par une méthode de régression. Ensuite, pour ces modèles, nous proposons des tests d’hypothèses sur σ. Enfin, dans les modèles plus généraux ci-dessus nous proposons des estimateurs fonctionnels pour la fonction σ(·) dont les propriétés sont obtenues via la convergence de fonctionnelles des accroissements doubles du mBf.

Let bH(t),t be the fractional brownian motion with parameter 0<H<1. When 1/2<H, we consider diffusion equations of the type

X(t)=c+0tσ(X(u))dbH(u)+0tμ(X(u))du.
In different particular models where σ(x)=σ or σ(x)=σx and μ(x)=μ or μ(x)=μx, we propose a central limit theorem for estimators of H and of σ based on regression methods. Then we give tests of the hypothesis on σ for these models. We also consider functional estimation on σ(·) in the above more general models based in the asymptotic behavior of functionals of the 2nd-order increments of the fBm.

DOI : 10.1214/07-AIHP105
Classification : 60F05, 60G15, 60G18, 60H10, 62F03, 62F12, 33C45
Mots-clés : central limit theorem, estimation, fractional brownian motion, gaussian processes, Hermite polynomials
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     title = {Estimation in models driven by fractional brownian motion},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {191--213},
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Berzin, Corinne; León, José R. Estimation in models driven by fractional brownian motion. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 2, pp. 191-213. doi : 10.1214/07-AIHP105. https://www.numdam.org/articles/10.1214/07-AIHP105/

[1] J.-M. Azaïs and M. Wschebor. Almost sure oscillation of certain random processes. Bernoulli 2 (1996) 257-270. | MR | Zbl

[2] C. Berzin and J. R. León. Convergence in fractional models and applications. Electron. J. Probab. 10 (2005) 326-370. | MR | Zbl

[3] C. Berzin and J. R. León. Estimating the Hurst parameter. Stat. Inference Stoch. Process. 10 (2007) 49-73. | MR | Zbl

[4] N. J. Cutland, P. E. Kopp and W. Willinger. Stock price returns and the Joseph effect: A fractional version of the Black-Scholes model. In Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1993) 327-351. Switzerland. | MR | Zbl

[5] L. Decreusefond and A. S. Üstünel. Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 (1999) 177-214. | MR | Zbl

[6] A. Gloter and M. Hoffmann. Stochastic volatility and fractional Brownian motion. Stochastic Process. Appl. 113 (2004) 143-172. | MR | Zbl

[7] F. Klingenhöfer and M. Zähle. Ordinary differential equations with fractal noise. Proc. Amer. Math. Soc. 127 (1999) 1021-1028. | MR | Zbl

[8] S. J. Lin. Stochastic analysis of fractional Brownian motions. Stochastics Stochastics Rep. 55 (1995) 121-140. | MR | Zbl

[9] T. Lyons. Differential equations driven by rough signals, I: An extension of an inequality of L. C. Young. Math. Res. Lett. 1 (1994) 451-464. | MR | Zbl

[10] B. B. Mandelbrot and J. W. Van Ness. Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 (1968) 422-437. | MR | Zbl

[11] D. Nualart and A. Răşcanu. Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2001) 55-81. | MR | Zbl

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