Can somebody help me with this exercise? Thanks a lot.
Let $X_1, X_2, \cdots$ be a sequence of non-negative independent random variables and consider $N(t) = \max\{n : X_1 + X_2 + \cdots + X_n \le t\}$. Define an appropriate filtration and show that $N(t) + 1$ is a stopping time with respect to the filtration.
For each $t\geqslant 0$, we have $$ \{N(t)+1\leqslant t\} = \{\max_n X_1+\cdots+X_{n-1}\leqslant t\} $$ which is $\mathcal F_{n-1}$-measurable, and is in fact predictable as well as a stopping time.