(I asked this question on MSE but I think it might have more success here)
Good day,
I was going over some exercises and I stumbled upon a question that, for its solution, requires me to find/simplify $$ \tilde{\Bbb{E}}[S_T|\mathcal{F}_t] $$ in terms of $S_t$ where $$ S_t=S_0Y_t+Y_t\int^t_0\frac{a}{Y_s}ds $$ $$ dY_t=rY_tdt+\sigma Y_td\tilde{W}_t$$ $$ \ Y_t=exp \left( \sigma\tilde{W}_t+(r-0.5\sigma^2)t \right) $$ $$ dS_t=rS_tdt+\sigma S_t d\tilde{W}_t +adt$$
$\tilde{\Bbb{P}}$ is the risk neutral measure.
$Y_t$ is a GBM and thus I think the first term is easy to deal with, but the 2nd one with the integral is a bit of a mystery to me. Do I have to take the $Y_T$ inside the integral and play with the exponential form of the GBM? Any help would be appreciated.
In essence, how do I find the following? $$ \tilde{\Bbb{E}}[Y_T\int^T_0\frac{a}{Y_s}ds|\mathcal{F}_t] $$