Vrftsjtryk

I really have problem in understanding stopping time. Let $(Omega ,mathcal F, (mathcal F_n),mathbb P)$ a filtred space. The definition is : $N$ is a stopping time if ${N=n}in mathcal F_n$ for all $ninmathbb N$. So indeed ${N=n}$ is measurable for all $n$, but what information it gives ?

For example, if $(X_n)$ is a matingale and $N$ is a stopping time, then $$mathbb E[X_0]=mathbb E[X_N],$$
in what the fact that $N$ is a stopping time is important ?

I have problem in understanding what stopping time represents.

Having struggled with this concept myself, I believe that the only way to fully grasp the concept of stopping time is to first understand the concepts of filtrations.

The space you mentioned is filtrated, i.e. there exists a family of sub $sigma$-algebras $mathcal{F}_n subsetmathcal{F}$ such that $mathcal{F_s} subset mathcal{F_t}$ for $s<t$.

All right, say we take some sub $sigma$-algebra of $mathcal{F}$, say $mathcal{F_t}$. What can we find inside the latter? Well, if the index of the filtration is referred to time, then $mathcal{F_t}$ contains the collection of all events observable up until time t.

So if the event is a stopping time by the definition that you gave, at any point in time you can tell whether the event has occurred or not (thanks to the information encoded in the filtration!).