1) Consider the Ito-integral:
$S_t = \int_{0}^{t}f(s)dW_S$, where $f$ is a Borel bounded function and $W$ is a Brownian motion.
Is the $\sigma$-algebra, $\sigma(S_t) = \sigma(W_s, s\leq t)?$ For sure it is obvious that the inclusion $\subset$ is true by definition of stochastic integral (limit of sums of Brownians). Is the opposite inclusion true?
2) Consider now the case of the Ito-Integral,
$S_t = \int_{0}^{t}f(V_s)dW_S$, with $V$-process satisfying the SDE:
$dV_s = -kV_sdt + \eta dW^V$, with $k, \eta$ constants and $W^V$ correlated to $W$ with correlation $\rho$.
In that case what is the $\sigma$-algebra, $\sigma(S_t)?$
Thanks!