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I'm trying to understand the proof by Itrel Monroe to the existence of minimal stopping times, in particular the following proposition from "On embedding right continuous martingales in Brownian Motion". Proposition 2

I struggle to understand a few points:

  1. Monroe consider a partial order "$\leq$" on the set of stopping times having the same law; it is a pointwise or almost sure inequality? I'm used to a.s. relations between random variables but here it isn't specified so I'm not sure.
  2. The sequence of stopping times $T_n$ is said to converge to a stopping time $T$ given that the sequence is decreasing; I presume pointwise, is it correct? Again I'm not sure.
  3. After saying that the $T_n \to T$ Monroe states that then $W_{T_n} \to W_T$ where $W_t$ is a Brownian motion. Here it seems implicit that there exists a result which guarantees some kind of convergence (what kind?) between stopped processes when the stopping times converges. I have not been able to find something along this line online.

Thank you in advance. It's my first question and English is not my native language so I apologize if I did something wrong.

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1 Answer 1

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1. Yes, the ordering here is the pointwise ordering: $T\le S$ means $T(\omega)\le S(\omega)$ for all $\omega$.

2. The extracted chain $\{T_n\}$ decreases pointwise, and the pointwise limit is $T$, another stopping time.

3. Because $\lim_nT_n(\omega)=T(\omega)$ for each $\omega$ and $t\mapsto W_t(\omega)$ is continuous for each $\omega$, you have $\lim_nW_{T_n}(\omega)=\lim_n W_{T_n(\omega)}(\omega)=W_{T(\omega)}(\omega) = W_T(\omega)$ for each $\omega$.

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