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On the Clark-Ocone theorem for fractional Brownian motions with Hurst parameter bigger than a half. (English) Zbl 1043.60027

It is known, that fractional analogues of the result by K. Aase, B. Øksendal, N. Privault and J. Ubøe [Finance Stoch. 4, No. 4, 465–496 (2000; Zbl 0963.60065)] on Clark-Ocone formula, obtained by Y. Hu and B. Øksendal [Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6, No. 1, 1–32 (2003; Zbl 1045.60072)] (\(1/2<H<1\)) and P. J. Elliott and J. van der Hoek [Math. Finance 13, No. 2, 301–330 (2003; Zbl 1069.91047)] (\(0<H<1\)), rely on the notion of quasi-conditional expectation, which is defined in terms of multiple fractional Wiener integrals (MFWI).
The present paper proves, that even if a square integrable random variable \(F\) has an expansion in terms of MFWI, its quasi-conditional expectation need not exist as a square integrable random variable, since the multiplication with an indicator function does not decrease the fractional norm, when \(1/2<H<1.\) Also, it is stated that the fractional Clark-Ocone derivative does not exist on a set of positive Lebesgue measures as square integrable random variable even for very regular fractional Malliavin differentiable random variables. Finally, a new version of the fractional Clark-Ocone formula for Hurst parameter \(1/2<H<1\) and an appropriate class of random variables are proved.

MSC:

60G15 Gaussian processes
60H05 Stochastic integrals
60H07 Stochastic calculus of variations and the Malliavin calculus
60H40 White noise theory
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