The quadratic variation for mixed-fractional Brownian motion.

Let ${W}=\lambda B+\nu B^{H}$ be a mixed-fractional Brownian motion with Hurst index $0< H<\frac{1}{2}$ and $\lambda,\nu\neq0$ . In this paper we study the quadratic covariation $[f({W}),{W}]^{(H)}$ defined by in probability, where f is a Borel function and $\eta_{s}=\lambda^{2}s+\nu^{2}s^{2H}$ . For some suitable function f we show that the quadratic covariation exists in $L^{2}(\Omega)$ and the Itô formula holds for all absolutely continuous function F with $F'=f$ , where the integral is the Skorohod integral with respect to W. [ABSTRACT FROM AUTHOR]