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The rough path associated to the multidimensional analytic fBm with any Hurst parameter. (English) Zbl 1220.60022

Summary: We consider a complex-valued \(d\)-dimensional fractional Brownian motion defined on the closure of the complex upper half-plane, called analytic fractional Brownian motion and denoted by \(\Gamma \). This process has been introduced by J. Unterberger [Ann. Probab. 37, No. 2, 565–614 (2009; Zbl 1172.60007)], and both its real and imaginary parts, restricted to the real axis, are usual fractional Brownian motions. The current note is devoted to prove that a rough path based on \(\Gamma \) can be constructed for any value of the Hurst parameter in \((0, 1/2)\). We also show how to solve differential equations driven by \(\Gamma \) in a neighborhood of 0 of the complex upper half-plane, by means of elementary arguments.

MSC:

60G22 Fractional processes, including fractional Brownian motion
60H05 Stochastic integrals
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)

Citations:

Zbl 1172.60007
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References:

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