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 $ H\in (0,1)$ is a centered Gaussian process $ B=\left\{ B_{t},t\geq 0\right\} $ with the covariance function.

$\displaystyle \ E(B_{t}B_{s})=\frac{1}{2}\left( s^{2H}+t^{2H}-\vert t-s\vert^{2H}\right) .$ (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 $ H>\frac{1}{3}$:

$\displaystyle \vert B_{t}-a\vert=\vert a\vert+\int_{0}^{t}\mathrm{sign}(B_{s}-a)dB_{s}+L_{t}^{a}.$ (2)

The stochastic integral appearing in this formula coincides the divergence operator with respect to the fBm, and $ L_{t}^{a}$ is the density of the occupation measure $ \Gamma \mapsto 2H\int_{0}^{t}1_{\Gamma }(B_{s})s^{2H-1}ds $. This result extends the classical Tanaka's formula for the Wiener process ( $ H=\frac{1}{2}$), where $ L_{t}^{a}$ 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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