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The Brownian motion has the following (Levy-Ciesielski?) construction via Hilbert space isomorphisms:

  1. Let $\{ Z_i \}_{i \in \mathbb{Z}}$ be i.i.d. $N(0,1)$ random variables defined on $(\Omega, \mathcal{F}, P)$.

  2. Let $\{ \phi_i \}_{i \in \mathbb{Z}}$ be an orthonormal basis for $L^2[0, 1]$.

  3. The map $\phi_i \mapsto Z_i $ extends to an isometry $B$ from $L^2[0, 1]$ to the subspace of $L^2(\Omega)$ generated by $\{ Z_i \}_{i \in \mathbb{Z}}$.

  4. The process defined by $B_t(\omega) = B(1_{[0,t]})(\omega)$ is a version of Brownian motion, and $\int_0 ^1 \phi_i dB_t = Z_i$ in the Ito sense.

Does this extend in some way to the fractional Brownian motion?

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    $\begingroup$ Initially asked on SE. $\endgroup$
    – Michael
    Commented Aug 18, 2014 at 18:58

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Yes, similar constructions essentially work for the fractional Brownian motion. The space $L^2[0,1]$ needs to be replaced by the space $\mathcal{H}$ which is the closure of the linear span of space of indicators $1_{[s,t]}$ with respect to the inner product

$ \langle 1_{[0,s]}, 1_{[0,t]} \rangle_\mathcal{H}=R(s,t)$

where $R(s,t)$ is the covariance function of the fractional Brownian motion. Details may be found in

Stochastic calculus for the fractional Brownian motion

which also contains relevant references.

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  • $\begingroup$ Thank you. What about the choice of basis? Can they be chosen in such a way that their integrals against fBm are, say, approximately uncorrelated? $\endgroup$
    – Michael
    Commented Aug 19, 2014 at 3:47
  • $\begingroup$ Yes, if $\phi_i$ is an orthonormal basis of $\mathcal{H}$, then $\mathbb{E}(B(\phi_i)B(\phi_j))=\delta_{ij}$. So the $B(\phi_i)$'s are uncorrelated. $\endgroup$
    – Fabrice Baudoin
    Commented Aug 19, 2014 at 14:30
  • $\begingroup$ I seem to be missing something. So the construction maps $1_{[s,t]}$ to $B^H_t - B^H_s$. Extending to the closure, for $f \in \mathcal{H}$, $\int f dB^H_t$ is defined to be the image of $f$ under this map. If we have an ONB $\{ \phi_i\}$ of $L^2[0,1]$ that happen to also be in $\mathcal{H}$, how do we know $\{ \int \phi_i dB^H_t \}$ are uncorrelated? Thanks for your patience. $\endgroup$
    – Michael
    Commented Aug 19, 2014 at 20:37

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