m order integrals and generalized Ito's formula the case of a fractional Brownian motion with any

m-order integrals and generalized Ito's formula; the case of a fractional Brownian motion with any Hurst index Mihai GRADINARU? ,(1), Ivan NOURDIN(1), Francesco RUSSO(2) and Pierre VALLOIS(1) (1) Universite Henri Poincare, Institut de Mathematiques Elie Cartan, B.P. 239, F - 54506 Vandœuvre-les-Nancy Cedex (2) Universite Paris 13, Institut Galilee, Mathematiques, 99, avenue J.B. Clement, F - 93430 Villetaneuse Cedex Abstract: Given an integer m, a probability measure ? on [0, 1], a process X and a real function g, we define the m-order ?-integral having as integrator X and as integrand g(X). In the case of the fractional Brownian motion BH , for any locally bounded function g, the corresponding integral vanishes for all odd indices m > 12H and any symmetric ?. One consequence is an Ito-Stratonovich type expansion for the fractional Brownian motion with arbitrary Hurst index H ?]0, 1[. On the other hand we show that the classical Ito-Stratonovich formula holds if and only if H > 16 . Key words and phrases: m-order integral, Ito's formula, fractional Brownian motion. 2000 Mathematics Subject Classification: 60H05, 60G15, 60G18.

• path approach


• symmetric integral


• odd indices


• hurst index


• has been


• ?bhs does


• when ?


• integral ∫