Some Analysis For Sample Paths Of Weighted Fractional Brownian Motions

In this article, we obtain the smoothness for the collision local time of fractional Brownian motion(fBm) and weighted fractional Brownian motion(wfBm), and also introduce the Ito and Tanaka formula of wfBm.We first consider fBm, a special case of wfBm. Recall that fBm with Hurst index H∈(0,1) is a Gauss process BH={BtH,t≥0} such that(i) B0H=0,(ii) EBtH=0,t≥0,(iii) E[BtH BsH=1/2(t2H+s2H-|t-s|2H), s,t≥0.Let BHi={BtHi,t≥0}, Hi∈(0,1), i= 1,2 be two independent fractional Brownian motion. The so-called collision local time is formally defined as whereδdenotes the Dirac delta function.In this artical, we will prove that lt is smooth in the sense of the Meyer-Watanabe if and only if min{H1,H2}<1/3.Besides, we consider the general wfBm Ba,b={Bta,b, t≥0} with covariance func-tionSimilarly as in the case of fBm, we can obtain the existence and the smoothness for the collision local time of wfBm. We derive the Ito and Tanaka formula of wfBm. Let f∈C2(R), Assume a≥-1/2, b≥0, then where∫0·f'(Bsa,b)δBsa,b is Skorohod Integral, and B(·,·) denotes the Beta function.The Ito formula above is also extended to the d-dimensional wfBm. In the end, we prove Tanaka formula where-1/2≤a<2, b>0.