I borrowed some lecture notes on stochastic calculus, which contained the following exercise:
Let $(X_n)_{n>0}$ be a sequence of random variables with $X_n: \Omega \to [0,\infty)$. We set $S_n= \sum_{i=1}^{n}X_i$, $S_0=0$ and $\mathcal{F_n}=\sigma(X_1,\dots,X_n)$. Let $a>0$ and set $L=\sup\{n \in \mathbb{N}: a \ge S_n\}$.
Is $τ=L+1$ a stopping time in respect to the filtration set above?
Well, I kinda feel that $τ$ is a stopping time, since $\{n-1\ge L\}$ can be translated as the intersection of $\{a \ge S_i\}$ for $i=1,\dots n-1$ which is $\mathcal{F_n}$-measurable for each $n$. But, if what i said above is right, then isnt $L$ a stopping time as well? Any hint that can help me clear my thoughts and find the solution would be appreciated.
$L+1=\inf\{n\mid S_n\gt a\}$ is a stopping time since, for every $n$, $[L+1\leqslant n]=[S_n\gt a]\in\mathcal F_n$.
On the other hand, $[L\leqslant n]=[S_{n+1}\gt a]$ is not in $\mathcal F_n$, hence $L$ is not a stopping time, in general.