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The strong Markov property is often formulated as $$P[\theta _{\tau}X\in A\mid \mathscr F_{\tau}]\overset{\text {a.s on }\left\{ \tau <\infty \right\} }{=}P_{X_\tau}(X\in A)$$

  • What exactly does "on $\left\{ \tau <\infty \right\}$" mean? Intuitively it seems we're conditioning on this event, but I'm unsure.
  • Is there a way to restate the strong Markov property as a conditional independence?

Edit: Since, as Did points out, the questions are unrelated, I'll ask the second one separately.

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Two unrelated questions, for the first one, the assertion that "$X=Y$ almost surely on $A$" means that the event $[X\ne Y]\cap A$ has probability zero.

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