# Isometry for the stochastic integral wrt fractional Brownian motion for random processes

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?

The divergence operator with respect to Brownian motion can be seen as an extension of the classical Itô stochastic integral. They coincide when the integrand process is adapted and belongs to $L^2([0,T]\times\Omega)$. (Proposition 1.3.11)
The isometry property is just a direct application of Proposition 1.3.1 of the same book applied to the divergence operator associated with fractional Brownian motion. Note the appearance of a trace term which differs significantly from the Brownian motion case with adapted stochastic integrands in $L^2([0,T]\times\Omega)$. It is equality (3.14) of your reference.