⧼exchistory⧽
2 exercise(s) shown, 0 hidden
BBy Bot
May 08'24
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The following exercises are important.
- Show that
[[math]] \F_T=\left\{\bigcup_{n\in\bar\N}A_n\cap \{T=n\}\big| A_\infty\in\F,A_n\in\F_n\right\}. [[/math]]
- Show that a r.v. [math]L[/math] with values in [math]\bar \N[/math] is a stopping time if and only if [math]\left(\one_{\{L\leq n\}}\right)_{n\geq 0}[/math] is [math](\F_n)[/math]-adapted and for the case it's a stopping time, we get
[[math]] L=\inf\{n\geq 0\mid \one_{\{ L\leq n\}}=1\}. [[/math]]
BBy Bot
May 08'24
[math]
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Let [math]T[/math] be a stopping time and [math]\Lambda\in\F_T[/math]. Define
[[math]]
T_\Lambda(\omega)=\begin{cases}T(\omega)&\text{if $\omega\in\Lambda$}\\ \infty&\text{if $\omega\not\in\Lambda$}\end{cases}
[[/math]]
Prove that [math]T_\Lambda[/math] is a stopping time.