The Property Of Mixed Fractional Brownian Motion And The Application In Finance

Let B^H = (B_t^H),t≥0 be a fractional Brownian motion (fBm) with Hurst pa-rameter H∈(0,1). Namely, (B_t^H,t≥0) is a centered Gaussian process whose E [B_t^H] =0, t≥0 and covariance is given byIn this article, we consider some aspects of mixed fractional Brownian motion M H= {M_t^H(a,b),t≥0}. By mixed fractional Brownian motion our meaning is a linear combination of independent fractional Brownian motions (see [15, 72]). Here we only consider a linear combination comprised by a fractional Brownian motion B^H with Hurst parameter H∈(0,1) and an independent Brownian motion B. That is,M_t^H(a, b) = aB_t^H + bB_t, t≥0,where a, b∈R.Firstly, for (?)≤H < 1 we consider the integral with respect to the weighted local time (?)ds of mixed fractional Brownian motionM^H(a,b)where f is a determinate function. By using the integral we give the characteristic for the quadratic covariation [f(M^H), M^H] of f(M^H) and M^H as follows[f(M^H),M^H]_t =(?)(dx,t). For the absolutely continuity functionwe obtain a general It(?) formula where the integral∫₀^TX_sdB_s^H is a type of Wick-It(?) integral, and the above also seems as a general It(?) formula for fractional Brownian motion. But actually the result for the fractional Brownian motion is not true since whose its quadratic covariation is zero.Next, we consider the self-intersection local time of mixed fractional Brownian motion M^H(a, b) of dimension d≥2 :and the collision local time of two independent mixed fractional Brownian motions of dimension 1 M^H₁(a₁,b₁) and M^H₂(a₂,b₂)We prove the existence and smoothness (in the situation of Meyer-Watanabe) of the two random variables L and l_T.Finally, we consider an application about mixed fractional Brownian motion, the financial market driven by the mixed fractional Brownian motion when (?)≤H < 1:where the integral∫₀^TX_sdB_s^H is a type of Wick-It(?) integral. We prove the market is complete and finally get a mixed fractional risk neutral pricing formula of European option.