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Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than 1/2. (English) Zbl 1028.60047
Stochastic integral with respect to fractional Brownian motion $$B^H_t$$ whose Hurst parameter $$H$$ is lesser than 1/2 is introduced. Integrability conditions are given by means of the Malliavin calculus and the notion of fractional derivative. Continuity properties of the indefinite integral are also studied and a maximal inequality is derived. The main idea of the approach utilizes the representation $B_t^H =\frac {1}{\Gamma (1-\alpha)}\Big [Z_t + \int _0^t(t-s)^{-\alpha } dW_s \Big ],$ where $$W_s$$ is a standard Brownian motion, $$\alpha =\frac {1}{2} - H$$ and $$Z_t=\int _{-\infty }^0[(t-s)^{-\alpha } - (-s)^{-\alpha }] dW_s$$. Since $$Z_t$$ is absolutely continuous, in order to develop stochastic calculus with respect to $$B^H_t$$ it suffices to consider the term $$B_t:= \int _0^t(t-s)^{-\alpha } dW_s$$. The stochastic integral $$\int _0^T \varphi (s)dB_s$$ is defined as the limit of integrals $$\int _0^t\varphi (s)dB^\varepsilon _s$$ as $$\varepsilon \to 0+$$ where $$B^\varepsilon _s := \int _0^t(t-s+\varepsilon)^{-\alpha }dW_s$$ are semimartingales.

##### MSC:
 60H05 Stochastic integrals 60G18 Self-similar stochastic processes
##### Keywords:
fractional Brownian motion; stochastic calculus
Full Text:
##### References:
 [1] Alòs, E.; Nualart, D., An extension of Itô’s formula for anticipating processes, J.. theoret probab., 11, 2, 493-514, (1998) · Zbl 0914.60018 [2] Carmona, P., Coutin, L., 1998. Stochastic integration with respect to fractional Brownian motion Preprint. · Zbl 0921.60067 [3] Dai, W.; Heyde, C.C., Itô’s formula with respect to fractional Brownian motion and its application, J. appl. math. stochastic anal., 9, 439-448, (1996) · Zbl 0867.60029 [4] Decreusefond, L.; Üstünel, A.S., Stochastic analysis of the fractional Brownian motion, Potential analysis, 10, 117-214, (1998) [5] Decreusefond, L.; Üstünel, A.S., Fractional Brownian motion: theory and applications, ESAIM: proc., 5, 75-86, (1998) · Zbl 0914.60019 [6] Duncan, T.E., Hu, Y., Pasik-Duncan, B., 1998. Stochastic calculus for fractional Brownian motion I, theory. Preprint. · Zbl 0947.60061 [7] Feyel, D., de la Pradelle, A., 1996. Fractional integrals and Brownian processes. Preprint, Université d’Évry. [8] Garsia, A.; Rodemish, E.; Rumsey, H., A real variable lemma and the continuity of paths of some Gaussian processes, Indiana univ. math. J., 20, 565-578, (1970) · Zbl 0252.60020 [9] Kleptsyna, M.L., Kloeden, P.E., Anh, V.V., 1996. Existence and uniqueness theorems for fBm stochastic differential equations. Preprint. · Zbl 0924.60042 [10] Lin, S.J., Stochastic analysis of fractional Brownian motions, Stochastics stochastics rep., 55, 121-140, (1995) · Zbl 0886.60076 [11] Mandelbrot, B.B.; Van Ness, J.W., Fractional Brownian motions, fractional noises and applications, SIAM review, 10, 4, 422-437, (1968) · Zbl 0179.47801 [12] Nualart, D., 1995. The Malliavin Calculus and Related Topics. Probability and Applications, Vol. 21, Springer, Berlin. · Zbl 0837.60050 [13] Nualart, D.; Pardoux, E., Stochastic calculus with anticipating integrands, Probab theory related fields, 78, 535-581, (1988) · Zbl 0629.60061 [14] Samko, S.G., Kilbas, A.A., Marichev, O.I., 1993. Fractional Integrals and Derivatives. Gordon and Breach Science, London. [15] Zähle, M., Integration with respect to fractal functions and stochastic calculus, Probab. theory related fields, 111, 3, 333-374, (1998) · Zbl 0918.60037
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