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In this article, we present the exact expression of the $L^2$-norm of theforward stochastic integral driven by the multi-dimensional fractional Brownianmotion with parameter $\frac{1}{2} < H < 1$. The class of integrands onlyrequires rather weak integrability conditions compatible w.r.t. a random finitemeasure whose density is expressed as a second-order polynomial of theunderlying driving Gaussian noise. A simple consequence of our results is theexact expression of the $L^2$-norm for the pathwise Young integral.

The $L^2$-norm of the forward stochastic integral w.r.t. Fractional Brownian motion $H > \frac{1}{2}$