Super basic question. I think I am doing this correctly, but just want a sanity check.
Say I have a stochastic process $r(t)$.
Say I have an equation
$$d(e^{\beta (t-s)}r(s))=\dots$$
where the $e^{\beta (t-s)}$ term is deterministic and $t\geq s > 0$.
Then say we want to integrate both sides from $s$ to $t$
$$\int_s^t{d(e^{\beta (t-u)}r(u))}=\dots$$
we then have
$$e^{\beta (t-t)}r(t)-e^{\beta (t-s)}r(s)=\dots$$
$$r(t)-e^{\beta (t-s)}r(s)=\dots$$
Please set me straight if I have this wrong. Thanks.