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My textbook introduces this deffinition. Let $(\Omega, \mathcal{F} , \mathcal{F}_t \mathbb{P}) $ be a filtered probability space. Let $\tau$ be a stopping time,

Then define $\mathcal{F}_{\tau} : = \{A \in \mathcal{F} : \forall t $ $ \{ \tau \leq t\} \cap A \in \mathcal{F}_t\}$

It states " $\mathcal{F}_{\tau}$ is the $\sigma$-algebra representing the information available up to the random time $\tau$"

I was having a hard time understanding the worded definition so decided to come up with a simple enough (discrete) example with coins.

Consider we toss two coins , then letting $\Omega = \{ H,T\}^2$ and $\mathcal{F}_0 = \{ \emptyset, \Omega\}$ , $\mathcal{F}_1 = \{\emptyset,\Omega , \{H,T \} \cup \{ H,H\} , \{T,H \} \cup \{ T,T\} \}$ and $\mathcal{F}_2 = \mathcal{P}(\Omega)$ we see that the $\mathcal{F}_t$ for $t = 0,1,2$ represent the natural filtration. Consider $\tau$ the hitting time of the first head tossed.

What would $\mathcal{F}_\tau$ be? What is an intuitive idea of what it contains? I do not understand the worded definition given to me.

I know so far that $\mathcal{F}_\tau$ contains $\emptyset, \Omega, TT , TH $ and I believe also $HT \cup HH$

OskarSzarowicz
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1 Answers1

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Let the probability space be $(\Omega,\mathscr{F},P)$. Let $(X_n)_{n \in \mathbb{N}}$ be IID Bernoulli with probability $p$. The filtration we consider is $\mathscr{F}_n:=\sigma(X_k,k\leq n)\subseteq \mathscr{F}$. Consider $\tau$ s.t. $\tau$ is a $\mathscr{F}_n$-stopping time. Note that $$\{X_{\tau}=1\}\cap \{\tau\leq n\}=\bigcup_{k \leq n}\{X_k=1\}\cap\{\tau=k\}\in \mathscr{F}_n,\forall n$$ or $$\{X_{\tau-1}=0\}\cap \{\tau\leq n\}=\bigcup_{k \leq n}\{X_{k-1}=0\}\cap\{\tau=k\}\in \mathscr{F}_n,\forall n$$ So for example $\{X_\tau=1\},\,\{X_{\tau-1}=0\}$ are sets in $\mathscr{F}_\tau$. Your case is $\tau=\inf\{n:X_n=1\}\wedge 2$.

Snoop
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