Some Processes Driven By Fractional Brownian Motion

Let B^H=(B_t^H),t≥0 be a fractional Brownian motion (fBm) with Hurst parameter 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 by In this paper, we consider some processes driven by fractional Brownian motion B^H.As the analogous of the self-attracting diffusions (see [19, 12]). we define the fractional self-attracting diffusion which is the unique strong solution of the following stochastic differential equation whereΦis a Lipschitz increasing function. In this paper, we consider only a particular linear case as follows: where, a＞0,v∈R. We show the process X_t^H converges to a random variable in L² almost surely as t→∞. We study the local time and weighted local time of this process and obtain Tanaka formula.Furthermore, we consider some processes associated with fractional Bessel process whereB^H = (B^H(1), B^H(2),...,B^H(d))is a d (d≥2)-dimensional fractional Brownian motion with Hurst parameter 0＜H＜1 and R=(B^H(1)²+B^H(2)²+…+B^H(d)²)^1/2 is the fractional Bessel process. In this paper, we obtain the local time of these processes and get the relationship between the weighted local time and the local time of fractional Bownian motion for d=1.Finally, for 2-dimensional linear fractional self-attracting diffusion, as a related problem, we show that the renormalized self-intersection local time exists in L² if 1/2＜H＜3/4.