In a classic framework $(\Omega,\mathcal A, P)$ with $B=(B_t)_{t\geq 0}$ a real Brownian motion and $(\mathcal F^B_t)_{t\geq 0}$ the canonical filtration tied to $B$.
Let this sequence of positives random variables $(T_n)_{n\in\mathbb{N}}$ (with $T_0=0$) and the recursive relationship : $$ \forall n\in\mathbb N,\ T_{n+1}=\inf\{t\geq T_n, \vert B_t-B_{T_n}\vert\geq 1\} $$ I want to prove that $(T_n)_{n\in\mathbb{N}}$ is a sequence of $(\mathcal F_t)$-stopping time.
The correction simply written : $$ \{T_{n+1}\leq s\} = \{T_{n}\leq s\} \cap \{\underset{T_n\leq u\leq s}{\sup} \vert B_u-B_{T_n}\vert\geq 1\} \in\mathcal F_s (*) $$
But I do not understand why this intersection.
My 2 observations about this problem :
- I think that we can prove that : $\{T_{n+1}\leq s\} = \{\underset{T_n\leq u\leq s}{\sup} \vert B_u-B_{T_n}\vert\geq 1\} (**)$
- $\{T_{n+1}\leq s\}\subseteq \{T_{n}\leq s\}$
(My observations are probably wrong, I do not know)
Questions :
If my observation (**) is true why the correction has written (*) ? (**) is not enough (to prove that it belongs to $\mathcal F_s$) ? Why it uses this intersection ? Is there a problem of open set ?
In the case that my observations are false, someone has understood (*) ?