Let us fix $(\Omega,\mathscr A,\Bbb P)$ a probability space. Let then $\Bbb F:=(\mathscr F_t)_{t\ge0}$ be a complete and right continuous filtration. Now if $B$ is an $\Bbb F$-standard Brownian motion, defining $$ M^2[0,T]:=\left\{X:\Omega\times[0,T]\to\Bbb R:X\;\mbox{prog. meas.}\;,\;\Bbb E\left[\int_0^TX_t^2dt\right]<+\infty\right\} $$ we can define the stochastic integral wrt to $B$ (the start by step processes and then extend on the closure which will turn out to be $M^2[0,T]$) and one of the most important properties is the Ito Isometry: $$ \Bbb E\left[\left(\int_0^TX_t\,dB_t\right)^2\right]=\Bbb E\left[\int_0^TX_t^2dt\right]. $$

Does exist an analogous Isometry for the stochastic integral wrt the *fractional Brownian motion* of Hurst index $H>1/2$?

I mean: considering a fractional BM $W^H=(W_t^H)_{t\ge0}$ with respect to $\Bbb F$, in section 3 of this paper by D. Nualart it can be found a quick introduction to the Stochastic Calculus wrt to FBM; it starts by considering *deterministic* processes, for which an Isometry property is stated (end of section 3.1.1).

Then (section 3.2) a few lines are spent to talk about integration of random processes, but nothing is said about isometry.

Does such a formula exist? If so, can you give me some references? Many thanks

**EDIT:** another useful question could be: does exist a way to express a stochastic integral wrt fractional BM in terms of a classical stochastic integral?