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The fBm Itô’s formula through analytic continuation. (English) Zbl 1008.60074

Summary: The fractional Brownian motion (fBm) can be extended to complex values of the parameter \(\alpha \) for \(\operatorname {Re}\alpha >{1\over 2}\). This is a useful tool. Indeed, the obtained process depends holomorphically on the parameter, so that many formulas, as Itô formula, can be extended by analytic continuation. For large values of \(\operatorname {Re}\alpha \), the stochastic calculus reduces to a deterministic one, so that formulas are very easy to prove. Hence they hold by analytic continuation for \(\operatorname {Re}\alpha \leq 1\), containing the classical case \(\alpha =1\).

MSC:

60H05 Stochastic integrals
60H07 Stochastic calculus of variations and the Malliavin calculus
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