If $X\_t$ is a semi-martingale, $\mathfrak{F}\_t$ is the $\sigma$-field generated by $X\_t$ and $L^2(Pred)$ is the set of all $\mathfrak{F}\_t$-predictible processes. Then is it true that:
$$
\mathfrak{G}\_t \triangleq \sigma\left(
H\cdot X\_t : H \in L^2(Pred)
\right)
=
\mathfrak{F}\_t?
$$



Writing $\sigma$-algebra in terms of predictble processes?
Yes, at least if $X\_0=0$. The containment $\mathfrak G\_t\subset\mathfrak F\_t$ is clear. In the other direction, is $0<s\le t$ then $X\_s=(H\cdot X)\_t$, where $H\_u(\omega) :=1\_{]0,s]}(u)$ is (trivially) predictable. This shows that $X\_s$ is $\mathfrak G\_t$-mesurable for each $s\in]0,t]$. 


