Recently I am working on the continuous-time martingale, I met such an result in S. N. Ethier and T. G. Kurtz. Markov Processes. Characterization and Convergence. John Wiley & Sons, New York, 1986. Theorem 2.13, p. 61: If $\left\{ X_{t},\mathscr{F}_{t}:0\le t<\infty\right\}$ is a right-continuous submartingale and $S,T$ are stopping times of $\left\{ \mathscr{F}_{t}\right\} $, then $ E\left[X_{\min\left\{ T,t\right\} }|\mathscr{F}_{S}\right]\geq X_{\min\left\{ T,S,t\right\} }$ for every $t \geq 0$.
However, I find it hard to comprehend its proof.
Here is Lemma 2.2.
My questions are:
(1) How do we get (2.39)? Lemma 2.2 only suits for finite-valued stopping times, but $\tau_i^{(n)}$ has countable values. And they also add "$\lor a$", I do not know how to show it.
(2) Similar question for equation (2.41).
(3) How to justify the Equation (2.42)? Although $X\left(\tau_{2}^{\left(n\right)}\land T\right)\vee\alpha$ is uniformly integrable and $X\left(\tau_{2}^{\left(n\right)}\land T\right)\vee\alpha\rightarrow X\left(\tau_{2}\land T\right)\vee\alpha$ by right-continuity, how can we get $E\left[X\left(\tau_{2}^{\left(n\right)}\land T\right)\vee\alpha|\mathscr{F}_{\tau_{1}}\right]\rightarrow E\left[X\left(\tau_{2}\land T\right)\vee\alpha|\mathscr{F}_{\tau_{1}}\right]$. The a.s. convergence and uniformly integrability may not implies the a.s. convergence of conditional expectation. See https://mathoverflow.net/questions/124589/uniformly-integrable-sequence-such-that-a-s-limit-and-conditional-expectation-d/124591
Thanks in advance!
Updates:
I solved the problem (i) and (ii) because I check the proof details of Lemma 2.2 and I found that using exactly the same proof, Lemma 2.2 can be extended to the case that "$T_{1}$ is a stopping time of $\left\{ \mathscr{F}_{t}\right\}$ assuming values in a countable set $\left\{ t_{1},t_{2},t_{3},\cdots\right\} \subset\left[0,\infty\right)$ with $t_{1}<t_{2}<\cdots$ and $T_{2}$ is a stopping time of $\left\{ \mathscr{F}_{t}\right\} $ assuming values in a finite set $\left\{ t_{1},t_{2},t_{3},\cdots,t_{m}\right\}$".
But I am still puzzled by question (3). Can anyone help me?
(1) Lemma 2.2 is being applied to $\tau_2^{(n)}\wedge T$ and $\tau_1^{(n)}\wedge \tau^{(n)}_2\wedge T$, both of which take only finitely many values.
(2) Ditto.