×

Generalized covariations, local time and Stratonovich Itô’s formula for fractional Brownian motion with Hurst index \(H\geq\frac 1 4\). (English) Zbl 1059.60067

The paper is devoted to generalized covariation processes and an Itô formula related to the fractional Brownian motion. The paper follows “almost pathwise calculus techniques” developed by Russo and Vallois, and it reaches the \(H=\frac{1}{4}\) barrier, developing very detailed Gaussian calculations. One motivation of this paper is to prove an Itô-Stratonovich formula for the fractional Brownian motion with \(H\geq\frac{1}{4}\). Such a process has, in some sense, a finite 4-variation and a finite pathwise \(p\)-variation for \(p>4\). It was even proved that the cubic variation is, in some sense, zero, when the Hurst index is bigger than \(\frac{1}{6}\). The main achievement is the proof of the existence of the 4-covariation \([g(B^H),B^H,B^H,B^H]\) for \(H\geq\frac{1}{4}\), \(g\) being locally bounded. Moreover, it is proved that this covariation is Hölder continuous with parameter strictly smaller than \(\frac{1}{4}\). The result provides, as an applications, the Itô-Stratonovich formula for \(f(B^H)\), \(f\) being of class \(C^4\) and a generalized Bouleau-Yor formula for fractional Brownian motion. Some results for local time are also obtained. The technique used here is a “pedestrian” but accurate exploitation of the Gaussian feature of fractional Brownian motion.

MSC:

60H05 Stochastic integrals
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H20 Stochastic integral equations
60G15 Gaussian processes
60G48 Generalizations of martingales
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] Alos, E., Léon, J. L. and Nualart, D. (2001). Stratonovich calculus for fractional Brownian motion with Hurst parameter less than \(\frac12\). Taiwanese J. Math. \(\mathbf 5\) 609–632. · Zbl 0989.60054
[2] Alos, E., Mazet, O. E. and Nualart, D. (1999). Stochastic calculus with respect to fractional Brownian motion with Hurst parameter less than \(\frac12\). Stochastic Process. Appl. 86 121–139. · Zbl 1028.60047
[3] Berman, S. (1973). Local non-determinism and local times of Gaussian processes. Indiana Univ. Math. J. 23 69–94. · Zbl 0264.60024
[4] Bertoin, J. (1986). Les processus de Dirichlet en tant qu’espace de Banach. Stochastics 18 155–168. · Zbl 0602.60069
[5] Bouleau, N. and Yor, M. (1981). Sur la variation quadratique des temps locaux de certaines semimartingales. C. R. Acad. Sci. Paris Sér. I Math. 292 491–494. · Zbl 0476.60046
[6] Carmona, P. and Coutin, L. (2000). Integrales stochastiques pour le mouvement brownien fractionnaire. C. R. Acad. Sci. Paris Sér. I Math. 330 213–236. · Zbl 0951.60042
[7] Cheridito, P. (2000). Regularizing fractional Brownian motion with a view towards stock price modelling. Ph.D. dissertation, ETH, Zurich.
[8] Coutin, L. and Qian, Z. (2000). Stochastic differential equations for fractional Brownian motions. C. R. Acad. Sci. Paris Sér. I Math. 330 1–6. · Zbl 0981.60040
[9] Coutin, L., Nualart, D. and Tudor, C. A. (2001). Tanaka formula for the fractional Brownian motion. Stochastic Process. Appl. 94 301–315. · Zbl 1053.60055
[10] Dai, W. and Heide, C. C. (1996). Itô formula with respect to fractional Brownian motion and its application. J. Appl. Math. Stochastic Anal. 9 439–448. · Zbl 0867.60029
[11] Decreusefond, L. and Ustunel, A. S. (1998). Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 177–214. · Zbl 0924.60034
[12] Delgado, R. and Jolis, M. (2000). On a Ogawa-type integral with application to the fractional Brownian motion. Stochastic Anal. Appl. 18 617–634. · Zbl 0981.60055
[13] Dudley, R. M. and Norvaisa, R. (1999). Differentiability of Six Operators on Nonsmooth Functions and \(p\)-Variation . Lecture Notes in Math. 1703 . Springer, Berlin. · Zbl 0973.46033
[14] Duncan, T. E., Hu, Y. and Pasik-Duncan, B. (2000). Stochastic calculus for fractional Brownian motion. I. Theory. SIAM J. Control Optim. 38 582–612. · Zbl 0947.60061
[15] Errami, M. and Russo, F. (1998). Covariation de convolutions de martingales. C. R. Acad. Sci. Paris Sér. I Math. 326 601–609. · Zbl 0917.60054
[16] Errami, M. and Russo, F. (2003). \(n\)-covariation, generalized Dirichlet processes and calculus with respect to finite cubic variation process. Stochastic Process. Appl. 104 259–299. · Zbl 1075.60531
[17] Feyel, D. and De La Pradelle, A. (1999). On fractional Brownian processes. Potential Anal. 10 273–288. · Zbl 0944.60045
[18] Föllmer, H. (1981). Calcul d’Itô sans probabilités. Seminar on Probability XV Lecture Notes in Math. 850 143–150. Springer, Berlin. · Zbl 0461.60074
[19] Föllmer, H., Protter, P. and Shiryaev, A. N. (1995). Quadratic covariation and an extension of Itô’s formula. Bernoulli 1 149–169. · Zbl 0851.60048
[20] Geman, D. and Horowitz, J. (1980). Occupation densities. Ann. Probab. 10 1–67. JSTOR: · Zbl 0499.60081
[21] Hu, Y. Z., Oksendal, B. and Zhang, T. S. (2000). Stochastic partial differential equations driven by multiparameter fractional white noise. In Stochastic Processes. Physics and Geometry : New Interplays (F. Gesztesy et al., eds.) 2 327–337. Amer. Math. Soc., Providence, RI. · Zbl 0982.60054
[22] Klingenhöfer, F. and Zähle, M. (1999). Ordinary differential equations with fractal noise. Proc. Amer. Math. Soc. 127 1021–1028. · Zbl 0915.34054
[23] Léandre, R. (2000). Stochastic Wess–Zumino–Novikov–Witten model on the sphere. Prépublication 2000-41, IECN.
[24] Lin, S. J. (1995). Stochastic analysis of fractional Brownian motion. Stochastics Stochastics Rep. 55 121–140. · Zbl 0886.60076
[25] Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 215–310. · Zbl 0923.34056
[26] Lyons, T. J. and Qian, Z. M. (1996). Calculus for multiplicative functionals, Itô’s formula and differential equations. In Itô’s Stochastic Calculus and Probability (M. Fukushima, ed.) 233–250. Springer, Berlin. · Zbl 0862.60043
[27] Lyons, T. J. and Zhang, T. S. (1994). Decomposition of Dirichlet processes and its applications. Ann. Probab. 22 494–524. JSTOR: · Zbl 0804.60044
[28] Lyons, T. J. and Zheng, W. (1988). A crossing estimate for the canonical process on a Dirichlet space and tightness result. Astérisque \(\mathbf 157/158\) 249–271. · Zbl 0654.60059
[29] Mémin, J., Mishura, Y. and Valkeila, E. (2001). Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion. Statist. Probab. Lett. 51 197–206. · Zbl 0983.60052
[30] Mishura, Yu. and Valkeila, E. (2000). An isometric approach to generalized stochastic integrals. J. Theoret. Probab. 13 673–693. · Zbl 0965.60054
[31] Pipiras, V. and Taqqu, M. S. (2000). Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118 251–291. · Zbl 0970.60058
[32] Protter, P. (1990). Stochastic Integration and Differential Equations. A New Approach. Springer, Berlin. · Zbl 0694.60047
[33] Revuz, D. and Yor, M. (1994). Continuous Martingales and Brownian Motion. Springer, Berlin. · Zbl 0804.60001
[34] Rogers, Ch. (1997). Arbitrage from fractional Brownian motion. Math. Finance 7 95–105. · Zbl 0884.90045
[35] Russo, F. and Vallois, P. (1991). Intégrales progressive, rétrograde et symétrique de processus non-adaptés. C. R. Acad. Sci. Paris Sér. I Math. 312 615–618. · Zbl 0723.60058
[36] Russo, F. and Vallois, P. (1993). Forward, backward and symmetric stochastic integration. Probab. Theory Related Fields 97 403–421. · Zbl 0792.60046
[37] Russo, F. and Vallois, P. (1995). The generalized covariation process and Itô formula. Stochastic Process. Appl. 59 81–104. · Zbl 0840.60052
[38] Russo, F. and Vallois, P. (1996). Itô formula for \(C^1\)-functions of semimartingales. Probab. Theory Related Fields 104 27–41. · Zbl 0838.60045
[39] Russo, F. and Vallois, P. (2000). Stochastic calculus with respect to a finite quadratic variation process. Stochastics Stochastics Rep. 70 1–40. · Zbl 0981.60053
[40] Sottinen, T. (2001). Fractional Brownian motion, random walks and binary market models. Finance Stoch. 5 343–355. · Zbl 0978.91037
[41] Wolf, J. (1997). An Itô formula for local Dirichlet processes. Stochastics Stochastics Rep. 62 103–115. · Zbl 0890.60044
[42] Young, L. C. (1936). An inequality of Hölder type, connected with Stieltjes integration. Acta Math. 67 251–282. · Zbl 0016.10404
[43] Zähle, M. (1998). Integration with respect to fractal functions and stochastic calculus. I. Probab. Theory Related. Fields 111 333–374. · Zbl 0918.60037
[44] Zähle, M. (2001). Integration with respect to fractal functions and stochastic calculus. II. Math. Nach. 225 145–183. · Zbl 0983.60054
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.