what is exactly the $\sigma -$algebra $\mathcal F_T=\{A\in \mathcal F\mid A\cap \{T\leq t\}\in \mathcal F_t\}$?

Asked Oct 16 '19 at 14:17

Active Oct 16 '19 at 14:17

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Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space, $\{\mathcal F_t\}$ a filtration and $T$ a stopping time. We denote $$\mathcal F_T=\{A\in \mathcal F\mid A\cap \{T\leq t\}\in \mathcal F_t\}.$$

What is exactly this $\sigma -$algebra ? What it represent concretely ?

asked Oct 16 '19 at 14:17

Herman

1

What do you mean by "concretely"? You might want to take a look at this question which is about the intuition behind $\mathcal{F}_T$. – saz Oct 16 '19 at 14:24
Here $A$ is an arbitrary element of $\mathcal F$ and if its intersection with ${T\leqslant t}$ is an element of $\mathcal F_t$, then $A\in \mathcal F_T$. – Math1000 Oct 16 '19 at 16:00
5

Does this answer your question? Verifying the interpretation of stopping times and stopping time $\sigma$-algebras – Marine Galantin Feb 21 '20 at 23:55

what is exactly the $\sigma -$algebra $\mathcal F_T=\{A\in \mathcal F\mid A\cap \{T\leq t\}\in \mathcal F_t\}$?

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