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Stochastic analysis of fractional Brownian motions. (English) Zbl 0886.60076
Summary: Some of the important ideas in ordinary stochastic analysis are applied to fractional Brownian motions (fBm’s). First we give a simple and elementary proof of the fact that any fBm has zero quadratic variation. This fact leads to the non-semimartingale structure of fBm’s. Another consequence is that we can integrate (in probability) the functionals of fBm’s with fBm differentials. With the same integrator, we then develop the \(L^2\) integration theory of bounded sure processes based on K. Bichteler’s integral extension theory. Finally, we investigate the corresponding stochastic differential equations with fractional Brownian noise.

MSC:
60J65 Brownian motion
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