Let $W$ be a Wiener process, $r,\sigma \in \mathbb{R}_+$ and $S(T) = S(t)e^{(r-\frac12 \sigma^2)(T-t) + \sigma(W(T)-W(t))}$.
I want to evaluate $A:=E[e^{- \frac12 \sigma^2 (T-t) - \sigma(W(T)-W(t)) } * \mathbf{1}_{\{S(T)>1.5\}}|\mathscr{F}_t].$
You will see that the left-hand factor of the expectation is an exponential martingale of the form $\mathcal{E}(-\sigma W)_t$.
Question: What can I do to progress further? I feel like I can't go:
$A = E[e^{- \frac12 \sigma^2 (T-t) - \sigma(W(T)-W(t)) }|\mathscr{F}_t] * E[\mathbf{1}_{\{S(T)>1.5\}}|\mathscr{F}_t]$
because $W(T)$ appears both inside the $e^{...}$ and inside the $\mathbf{1}_{\{...\}}$, making them dependent.