Tanaka formula for the fractional Brownian motion
Laure Coutin1) |
|
David Nualart2) |
|
Ciprian A. Tudor3) |
1) Laboratoire de Statistique et de Probabilités,
Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France.
2)
Facultat de Matemàtiques,
Universitat de Barcelona,
Gran Via 585, 08007, Barcelona, Spain.
3) Département de Mathématiques,
Université de La Rochelle,
Avenue Michel Crépau, 17042 La Rochelle Cedex 1, France.
Abstract:
The fractional Brownian motion (fBm) of Hurst parameter
is a
centered Gaussian process
with the
covariance function.
|
(1) |
There has been a recent development in the stochastic calculus with respect
to this process. Different approaches have been used to define stochastic
integrals and to establish change-of-variable formulas.
The purpose of this paper is to establish the following version of
Tanaka's formula for the fractional Brownian motion, assuming
:
|
(2) |
The stochastic integral appearing in this formula coincides the divergence
operator with respect to the fBm, and
is the density of the
occupation measure
. This result extends the classical Tanaka's formula for
the Wiener process (
), where
is the local time of
the Brownian motion, and the stochastic integral is an Itô integral.
Key words:
Fractional Brownian motion, local time, Tanaka formula.
Mathematics Subject Classification:
60G15, 60H07, 60H05, 60J65, 60F25.
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