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Let $\{B_t\}_{t\geq 0}$ be standard Brownian motion with respect to filtration $\{\mathcal{F}_t\}_{t\geq 0}$. Let $T$ be any stopping time. Is it true that $B_1-B_{T\wedge 1}$ independent of $\mathcal{F}_T$, or is it true that $\mathbb{E}[|B_1-B_{T\wedge 1}|\,|\,\mathcal{F}_T]=\mathbb{E}[|B_1-B_{T\wedge 1}|]$?

I am not sure whether $\{\mathcal{F}_t\}_{t\geq 0}$ is right continuous or not, but I know $B_t$ has continuous path and $B_{t\wedge 1}$ is a uniformly integrable martingale. When I search strong Markov property for Brownian motion, textbooks or other notes usually give me $B_{T+t}-B_T$ is independent of $\mathcal{F}_{T+}$ and they usually assume $P(T<\infty)>0$. I am not sure whether these regularity assumption is needed. Can anyone help?

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It is not true that $\mathbb{E}[|B_1-B_{T\wedge 1}|\,|\,\mathcal{F}_T]=\mathbb{E}[|B_1-B_{T\wedge 1}|]$? If this is true then (since $(T>1) \in\mathcal F_T$) we get $0=\mathbb{E} I_{T>1}[|B_1-B_{T\wedge 1}|=\mathbb{E}[|B_1-B_{T\wedge 1}|]P(T>1)$ which is clearly false.

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  • $\begingroup$ Thanks for your answer @KaviRamaMurthy. It seems we cannot directly remove $\mathcal{F}_T$, then do you think we can get rid of the conditional expectation by the following upper bounding process: $\mathbb{E}[|B_1-B_{T\wedge 1}|\,|\,\mathcal{F}_T]\leq 2\mathbb{E}[\sup_{t\in[0,1]}|B_t|\,|\,\mathcal{F}_T]=2\mathbb{E}[\sup_{t\in[0,1]}|B_t|]$? $\endgroup$
    – William
    Feb 15, 2022 at 5:15
  • $\begingroup$ The first inequality is fine but the last step is wrong. You cannot get rid of the conditioning. @William $\endgroup$
    – Kavi Rama Murthy
    Feb 15, 2022 at 5:20
  • $\begingroup$ Thanks again @KaviRamaMurthy, I want to do that because I am trying to upper bounding the term $\mathbb{E}[|B_1-B_{T\wedge 1}|\,|\,\mathcal{F}_T]$ by a constant. Currently I know $\mathbb{E}[|B_1-B_{T\wedge 1}|]$ and $\mathbb{E}[\sup_{t\in[0,1]}|B_t|]$ can be bounded above by a constant. If the conditional expectation cannot be avoided, do you know any approach that I could try? $\endgroup$
    – William
    Feb 15, 2022 at 5:32

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