Raw versus usual filtrations in the strong Markov theorem

The strong Markov theorem Strong Markov property of Brownian motion is stated for the usual augmentation $({\mathcal{F}}_t)$, not for the raw natural filtration $({\mathcal{F}}_t^0)$ Natural and usual augmented Brownian filtrations. Four distinctions matter and are fixed by that choice.

1. The theorem names the filtration it uses. Its hypothesis is that $\tau$ is a stopping time for $({\mathcal{F}}_t)$ with $\tau <\mathrm{\infty }$ almost surely; the conclusion is an identity of conditional expectations given ${\mathcal{F}}_{\tau }$. Neither the raw filtration nor the completed raw filtration is substituted for the usual one in the statement.
2. What the dyadic proof actually uses. The ceiling times ${\tau }_n=2^{-n}\lceil 2^n\tau \rceil$ are stopping times of the same filtration and satisfy ${\mathcal{F}}_{\tau }\subseteq {\mathcal{F}}_{{\tau }_n}$; the countably valued case applies the deterministic future-path theorem Future-path Markov property at the countably many values of ${\tau }_n$; and the passage to general $\tau$ uses path continuity and dominated convergence. Completion enters through the null event $\{\tau =\mathrm{\infty }\}$, on which $B_{\tau }$ is defined by a convention, and through the identification of conditional laws up to null sets.
3. Completion is not independence from arbitrary future information. The theorem asserts that the increment process $(B_{\tau +t}-B_{\tau }{)}_{t\ge 0}$ is independent of ${\mathcal{F}}_{\tau }$ and that the conditional law of the shifted future path is Wiener measure translated by $B_{\tau }$. For $\tau$ equal to a deterministic time $s>0$ this is not the claim that the future path $(B_{s+t}{)}_{t\ge 0}$ is independent of ${\mathcal{F}}_s$: its conditional law depends on the state $B_s$ through the translation, and only the increment process is independent of the past. No completion of the filtration removes that dependence, and none of the items on this page asserts it.
4. The stopping-time hypothesis is not decorative. For a random time that is not a stopping time the conclusion can fail outright; the companion example cex-strong-markov-fails-at-a-nonstopping-random-time on the companion examples page exhibits the last zero before a fixed time, where the post-time future has no zero in a right-neighbourhood and therefore cannot have the Wiener law.

The strict and non-strict forms of the stopping tests agree for the usual augmentation because it is right-continuous, while the raw statements on this page use the non-strict test directly; the ceiling identity $\{{\tau }_n\le t\}=\{\tau \le 2^{-n}\lfloor 2^{n}t\rfloor \}$ is a non-strict test and needs no right-continuity. AC is declared because the conditional-expectation interface and the ambient Brownian construction assume it.

• Reading order. The example items named by ID above are homed on later pages of the plan, so they are named rather than hyperlinked: a body link to later material must be declared as a forward reference, and Step-5b closure removes every such declaration. Rehoming those items to an earlier page (an owner-only reading-order change) would make the citations backward and restore the links.

Sousi, Sections 6.3-6.5, distinguishes the natural filtration from its right-continuous completion and states the strong Markov property for the latter; Durrett, Section 7.3, works throughout with the completed filtration. The counterexample on the companion page is oriented as a boundary for the stopping-time hypothesis, not used as a supplier anywhere in this pair.