I am currently working on an exercise on Poisson processes and I'm completely stuck. The question is as follows:
Suppose that you earn random iid amounts of money $X_i$ at times $T_i$, which occur according to a Poisson process $N(\cdot)$. It turns out that the richer you are the faster you spend money. Assume continuously at some rate $rx$ with $r>0$, for an account balance x. The bank account balance then follows the equation:
$$ B(t) = \sum_{i=1}^{N(t)} X_i \ e^{-r(t-T_i)} $$
Calculate the expected account balance at time $t$, i.e. $\mathbb{E}B(t)$
The hint is given to condition on $N(t)$ and $T_1,T_2,...,T_{N(t)}$.
I am absolutely clueless how to start. I figured that when I condition on $N(t) = n$ the jump times have joint density function $f(T_1,...,T_n) = n! \ t^{-n} \ \mathbb{1}_{(0 \leq T_1 \leq ... \leq T_n \leq t)}$