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 $(\alpha_n \mu)(t)$. Let $0=T_0^{(n)}<T_1^{(n)}<..$ jumps of $N^{(n)}$.
Let $M_n(t)=\sum_{i=1}^{N_t^{(n)}} \frac {1} {\alpha_n (T_i^{(n)})}$ and $M(t)=\int_0^t \mu(s)ds$
Calculate $E((Mn(t)-M(t))^2)$?
Before I showed that $M_n(t)$ was an unbiased estimator of M(t) But here I don't see how to solve the question
Can you help me?
Thank you