conditional expectation, exercice:difficult calculus Let $\mu,\alpha_n:\mathbb R^+\to \mathbb R$ continuous function with $\mu$ bounded function.
Let $N^{(n)}$ the trajectory of a Poisson process with intensity $\alpha_n \mu$.
Let $0=T_0^{(n)}<T_1^{(n)}<..$ jumps of $N^{(n)}$
Let $M_n(t)=\sum_{i=1}^{N_t^{(n)}} \frac {1} {\alpha_n (T_i^{(n)})}$
Calculate $E(M_n(t)|N_t^{(n)})$
My idea:
$E(M_n(t)|N_t^{(n)})=E(M_n(t)|N_t^{(n)}=u)=E\left(\sum_{i=1}^{u} \frac {1} { \alpha_n (T_i^{(n)})}\right)=\sum_{i=1}^{u}E\left(\frac {1} { \alpha_n (T_i^{(n)})}\right)$
But then I don't know. Can you help me?
Thank you

For a Poisson process the times between jumps, $T^{(n)}_{k}-T^{(n)}_{k-1}\ ,$ are independent and exponential distributed random variables. This implies that individual jump times are sums of independent exponential random variables, hence they are Gamma-distributed. $$T^{(n)}_{k} = \left(T^{(n)}_{k}-T^{(n)}_{k-1}\right) + \left(T^{(n)}_{k-1}-T^{(n)}_{k-2}\right) + \cdots + \left(T^{(n)}_{1}-T^{(n)}_{0}\right)\ ,$$ noting that $T^{(n)}_{0} = 0 \ .$
