Suppose we have an SDE, which is the Wiener process with drift
$dr_t=c dt+\sigma dB_t$, where $B_t$ is brownian
I want to find $\mathbb{E}[e^{-\int_0^t r_s ds} |r_t=r]$
so my approach is this : write the SDE as : $r_t-r_0=ct+\int\sigma dB_t$
Then I know $r_t$ is distributed as a normal. but then how can i get the distribution of $\int_0^t r_s ds$ and hence the expectation?
thanks