I'm trying to understand the proof of Corollary 8.7 of Chapter 4 in the book Markov Processes: Characterization and Convergence by Stewart N. Ethier, Thomas G. Kurtz. Here is the theorem and its proof:
And here are the relevant parts they refer to:
There are two things that I don't understand:
- Why do they use $\chi_{\left\{\:\tau_n\:>\:\color{red}{1}\:\right\}}$ (indicator function of $\left\{\tau_n>1\right\}$) in the definition of $\varphi_n$. Wouldn't it make more sense to use $\chi_{\left\{\:\tau_n\:>\:\color{red}{T}\:\right\}}$ instead?
- By definition, $f_n(Y_n(t))-\int_0^tg_n(Y_n(s))\:{\rm d}s$ is a martingale with respect to the filtration $\mathcal F^{Y_n}$ generated by $Y_n$. But why is $\xi_n(t)-\int_0^t\varphi_n(s)\:{\rm d}s$ still a $\mathcal F^{Y_n}$-martingale? Since they seem to conclude the proof with Theorem 8.2 (c), it seems like they are assuming this (since in Theorem 8.2, $(\xi_n,\varphi_n)\in\hat{\mathcal A}_n$)