I have a doubt about expected value of this stochastic process:
Let $X_n$ be a succession of iid gaussian standard r.v. $X_k$~$N(0,1)$ and let $S_n=\sum_{k=1}^n X_k$ and let $Z_n=e^{(-\frac{\sigma^2}2(n)+\sigma\sum_{k=1}^n X_k)}$.
Find $E[Z_{n+1}|F_n]$
I did this using the TOWIK property
$$E[Z_{n+1}|F_n]=e^{-\frac{\sigma^2}2(n+1)+\sigma\sum_{k=1}^n X_k}*E[e^{X_{n+1}}|F_n]$$
Now, my question is, can I write this?
$$e^{-\frac{\sigma^2}2(n+1)+\sigma\sum_{k=1}^n X_k}=e^{-\frac{\sigma^2}2(n+1)+\sigma*n X_k}$$
I thought this beacuse I already know all the value of the process for time $k_1,k_2,....,k_n$ so why should I keep an aleatory term in my equation like this $e^{\sigma\sum_{k=1}^n X_k}$, that is a random variable?
I thought that this term should be deterministic beacause I already know that value.
So, can I write $e^{\sigma\sum_{k=1}^n X_k}=e^{\sigma*n X_k}$ by taking out what is known from the expected value?
Thank You