The law of a stochastic integral with two independent fractional Brownian motions

Using the tools of the stochastic integration with respect to the fractional Brownian motion, we obtain the expression of the characteristic function of the random variable $\int_{0}^{1}B^{\alpha }_{s}dB^{H}_{s}$ where $B^{\alpha }$ and $B^{H}$ are two independent fractional Brownian motions with Hurst parameters $\alpha\in(0,1) $ and $H>\frac12$ respectively. The two-parameter case is also considered