If $T$ is a stopping time, what represent $\mathcal F_T$?

Question

Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space and let $T:\Omega \to \mathbb N$ a stoping time. Let $(X_n)_{n\geq 1}$ a stochastic process and $\mathcal F_n=\sigma (X_1,...,X_n)$, and $\mathcal F_\infty =\sigma (X_1,X_2,...)$.

I don't really understand what $$\mathcal F_T=\{A\cap \{T=n\}\in \mathcal F_n\mid A\in \mathcal F_\infty \}.$$ I saw on the internet that it's called the $\sigma -$algebra of previous event, but that doesn't really enlighten me.

Also, what would be $$\mathcal G_T=\{A\cap \{T=n\}\in \mathcal F_n\mid A\in \mathcal F\} \ \ ?$$ is $\mathcal G_T$ interesting ?

Answer 1

When you are studying a process $\{X_n\}$ you might as well assume that $\mathcal F =\mathcal F_{\infty}$. Sets in $\mathcal F$ which don't belong to $\mathcal F_{\infty}$ don't come into the picture in the analysis of $\{X_n\}$. Se we can take $\mathcal F_T=\mathcal G_T$.

$\mathcal F_T$ consists (intuitively) of events $A$ such that when you restrict your attention to $\{T=n\}$ the events is defined completely in terms of $X_1,X_2,...,X_n$ (and $X_k$ with $k >n$ are not involved).

Answer 2

Note that one can prove the equivalent following definition :

$$\mathcal F_T = \bigvee^\infty_0 \sigma(x_{T \wedge n }) $$ where $x_{T \wedge n } $ represents the process equal to $x_n$ when $n$ hasn't reached the stopping time.