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Integration questions related to fractional Brownian motion. (English) Zbl 0970.60058

The authors study integration with respect to fractional Brownian motion. Their aim is to characterize the linear span H B H of the fractional Brownian motion B H (t), t, H(0,1), in terms of some function space. This means that they study integration of deterministic functions. For H(0,1/2) it is possible to give a complete characterization of the space H B H as follows: Let

Λ H ={f:ϕL 2 ()s.t.f=I - 1/2-H ϕ},

where I - α is the fractional integral operator of order α. For f,gΛ H put

(f,g) Λ H =c H (ϕ f ,ϕ g ) L 2 () ·

Then the space Λ H is isometric to H B H . For H(1/2,1) the authors show that it is not possible to obtain such a characterization in terms of a function space.

The authors also study related problems in the spectral domain using the spectral representation of fractional Brownian motion in terms of a complex Gaussian measure. They consider the following function space

Λ ˜ H =f : f L 2 () , f ^ (x) 2 |x| 1-2H d x < ·

They show that the elementary functions are dense in Λ ˜ H , but this space is not complete, unless H=1/2. In addition, they show that Λ ˜ H Λ H , the inclusion is strict, when H1/2, and where the space Λ H for H(1/2,1) is defined as

Λ H =f : I - H-1/2 (f) (s) 2 d s < ·


MSC:
60H05Stochastic integrals
60G18Self-similar processes
26A33Fractional derivatives and integrals (real functions)