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: I know that: $$E(M_n(t)|N_t^{(n)})=\int...\int g(t_1,...,t_u)\sum_{i=1}^{u} \frac {1} {\alpha_n \left(t_i^{(n)}\right)}dt_1...dt_u$$ with:
$$g(t_1,...,t_u)=\frac{u!}{h(t)^u}(\alpha_n \mu)(t_1)...(\alpha_n \mu)(t_n)1_{0 \leq t_1<...<t_u \leq t}$$ and $h(t)=\int_0^t (\alpha_n \mu)(s)ds$
But have you got ideas to simply my expression?
Thank you