Let $B^{H_i,K_i}=\{B_t^{H_i,K_i},t\ge 0\}$, $i=1,2$ be two independent, $d$-dimensional bifractional Brownian motions with respective indices $H_i\in (0,1)$ and $K_i\in (0,1]$. Assume $d\ge 2$. One of the main motivations of this paper is to investigate smoothness of the collision local time $${\ell }_T={\int }_0^T\delta (B_s^{H_1,K_1}-B_s^{H_2,K_2})\mathrm{d}s,T>0,$$ where $\delta$ denotes the Dirac delta function. By an elementary method we show that ${\ell }_T$ is smooth in the sense of Meyer-Watanabe if and only if $min\{H_1{K}_1,H_2{K}_2\}<1/(d+2)$.

A stochastic affine evolution equation with bilinear noise term is studied, where the driving process is a real-valued fractional Brownian motion with Hurst parameter greater than $1/2$. Stochastic integration is understood in the Skorokhod sense. The existence and uniqueness of weak solution is proved and some results on the large time dynamics are obtained.

Fractional Brownian motion (fBm) is a centered self-similar Gaussian process with stationary increments, which depends on a parameter $H\in (0,1)$ called the Hurst index. In this conference we will survey some recent advances in the stochastic calculus with respect to fBm. In the particular case $H=1/2$, the process is an ordinary Brownian motion, but otherwise it is not a semimartingale and Itô calculus cannot be used. Different approaches have been introduced to construct stochastic integrals with respect to fBm:...