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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
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