Given a filtration, ${\mathcal{F}_t},t\in[0,\infty).$ Let $T_n$ be a sequence of stopping time that converges to $T$ and $T_n\le T_{n+1}.$ We have correpsonding $\sigma-$algebra, ${\mathcal{F}_{T_n}}$ and $\mathcal{F}_T.$
Now, denote $\mathcal{F}'=\sigma(\mathcal{F}_{T_n}:n=1,2,\cdots),$ i.e., the $\sigma-$algebra generated by all $\mathcal{F}_{T_n}$.
Q: Will $\mathcal{F}'=\mathcal{F}_T$ holds?
I believe the condition that the filtration is left continuous is needed, since one can take $T_n,T$ to be constant. Let's assume that.
My try:
- That $\mathcal{F}'\subset\mathcal{F}_T$ is trivial, since all $\mathcal{F}_{T_n}\subset\mathcal{F}_T$ by $T_n\le T$. We left to show $\mathcal{F}_T\subset\mathcal{F}'.$
- By definition, $A\in\mathcal{F}_T$ is equivalent to $A\cap\{T\le t\}\in\mathcal{F}_t$ for any $t$. How can one deduce from here that $A\in \mathcal{F}'.$ Got stuck here.
- In Approximation of a unbounded stopping time and convergence of respective $\sigma$-algebras, saz gives an approach for discrete time. For $A\in\mathcal{F}_T, A$ can be decomposed as, $$A = \cup_{n=1}^{\infty} (A\cap\{T\le n\})\cup(A\cap\{T=\infty\}) = \cup_{n=1}^{\infty} A_n\cup A_\infty.$$ Then show $A_n\in\mathcal{F}_{T_n}.$ We have $A_n\in\mathcal{F}_n, $ we require to show $A_n\cap\{T_n\le t\}\in\mathcal{F_t},$ this is so if $t\ge n$ since $\{T_n\le t\}\in\mathcal{F}_t$. But how about $t<n?$ I was stuck here.
Update: One can define, $\mathcal{F}_{S-}=$ the $\sigma-$algebra generated by $\mathcal{F}_{0+}=\cap_{s>0}\mathcal{F}_s$ and the sets $\{S>t\}\cap\mathcal{F_t}.$ then when $S=s$ is a constant, $\mathcal{F}_{S-}=\sigma(\mathcal{F}_u:u<s)=\mathcal{F}_{s-}.$ So this is simply the generalization of left limit. Then one can prove that, $$\mathcal{F}_{T-}=\mathcal{F}'.$$
So now, the question may become to show, $$ \mathcal{F}_{T-}=\mathcal{F}_{T} .$$
Any hint is appreciated!
The inclusion $\mathcal F'\subset \mathcal F_T$ may be strict.
Example: Let $(\Omega,\mathcal F,\Bbb P)$ be a probability space on which is defined a standard normal random variable $Z$. (So that $\Bbb P[Z=0]=0$.) Let $(\mathcal F_t)$ be the filtration generated by the process $X$ defined by $$ X_t=\cases{1,&$0\le t<1$,\cr Z,&$t\ge 1$.\cr} $$ For $T_n$ take the constant $1-1/n$, and then $T=1$. Yiu have $\mathcal F_{T_n}=\mathcal F' = \{\emptyset,\Omega\}$ for all $n$, but $\mathcal F_T=\sigma\{Z\}$. The key here is that $T$ is predictable, and "something new" happens at time $T$.