Filtration generated by process $S_t=S_0 \exp \left(\mu t-\frac{\sigma^2}{2}t+\sigma W_t\right)$

Question

$W_t$ – Wiener process and $(F_t^W)$ its natural filtration.

$$S_t=S_0 \exp \left(\mu t-\frac{\sigma^2}{2}t+\sigma W_t\right)$$

where $S_0,\sigma>0, \mu\in\mathbb{R}$, $(F_t^S)$ – natural filtration of process $(S_t)$. Prove that $F_t^S=F_t^W$ for every $t\ge0$

how to prove it?

$\sigma\left\{S_s:0\le s\le t\right\}=\sigma\left\{S_0 \exp \left(\mu s-\frac{\sigma^2}{2}s+\sigma W_s\right):0\le s\le t\right\}$ and what next?

Answer 1

Hint

$$W_t=\frac{1}{\sigma }\left( \ln(S_t)-\ln(S_0)-\left(\mu-\frac{\sigma ^2}{2} \right)t\right).$$