# stopping time definitions

• February 26th 2013, 05:12 AM
alphabeta89
stopping time definitions
Let $\tau$ a stopping time w.r.t. the filtration $(F_n)_{n\in{\mathbb{N}}}$. The stopped $\sigma$-field $F_\tau$ associated with $\tau$ is defined to be

$F_\tau:={\{A\in{F}:A\cap{\{\omega:\tau(\omega)=n\} \in{F_n} \text{ for all }n\geq {0}}\}$.

I know this definition is equivalent to

$F_\tau:={\{A\in{F}:A\cap{\{\omega:\tau(\omega)\leq {n}\}\in{F_n} \text{ for all }n\geq {0}}\}$.

But is it the same as

$F_\tau:={\{A\in{F}:A\cap{\{\omega:\tau(\omega)?
I have a feeling the answer is no, but I don't know how to start disproving it. Any help is appreciated! Thanks!:D