Q:

Adaptation of sum of arrival times of Poisson process

Let $\{N_t\}_{t\geq0}$ be a Poisson process and $\{F_t\}_{t\geq0}$ be its nautral filtration so that $\{N_t\}_{t\geq0}$ is adapted. $T_i$ be the $i$th arrival time of Poisson process of arrival rate $\lambda$, given $t>0$, prove if $\sum_{i=1}^{N_t} T_i$ is also adapted to $\{F_t\}_{t\geq0}$?

A:

Let $L_t=\sum\limits_{i=1}^{N_t}T_i$, then $L_t=\int\limits_0^t(N_t-N_s)\mathrm ds$. Since $(N_s)_{0\leqslant s\leqslant t}$ is $F_t$-measurable, this proves that $L_t$ is $F_t$-measurable.
To prove the key-formula used above, fubinize $L_t$, that is, using the identity $[s\lt T_i]=[N_s\lt i]$, write $L_t$ as
$$
L_t=\sum\limits_{i\geqslant1}\mathbf 1_{i\leqslant N_t}\int_0^\infty \mathbf 1_{s\lt T_i}\mathrm ds=\int_0^\infty \sum\limits_{i\geqslant1}\mathbf 1_{N_s\lt i\leqslant N_t}\mathrm ds=\int_0^\infty(N_t-N_s)^+\mathrm ds.
$$

