How to prove the exercise related to stopping time in stochastic process?

Question

N is stopping time and $Y_n \in \mathbf{F}_n$, prove that $Y_N \in \mathbf{F}_N$.

Answer 1

To prove $Y_N \in \mathcal F_N$, we need to show $\{ Y_N \le a \} \in \mathcal F_N$ for all $a \in \mathbb{R}$. From the definition of $\mathcal F_N$, this means showing $\{N \le n \} \cap \{ Y_N \le a \} \in \mathcal F_n$ for all $n \in \mathbb{N}$. We know $\{N \le n \} = \bigcup_{m=1}^n \{N = m\}$ so \begin{align*} \{N \le n \} \cap \{ Y_N \le a \} &= \bigcup_{m=1}^n( \{N = m\}\cap \{ Y_N \le a \}) = \bigcup_{m=1}^n( \{N = m\}\cap \{ Y_m \le a \}) \end{align*}

is the finite union of events in $\mathcal F_n$ and hence is in $\mathcal F_n$. We conclude $\{ Y_N \le a \} \in \mathcal F_N$ and hence $Y_N \in \mathcal F_N$.