Stopping time of $ \min \left\{ n \geq 1 : X_n = Y_n \right\}$

Question

Let $(\Omega, \mathcal{A},(\mathcal{F}_n)_{n≥1},\mathbb{P})\;$ be a given filtrated probability space, and $X = (X_n)_{n≥1}$ , $Y = (Y_n)_{n≥1}\;$ be two $(\mathcal{F}_n)_{n≥1}$-martingales. Set $$ \tau := \min \left\{ n \geq 1 : X_n = Y_n \right\} $$ How we can show that $\tau\;$ is an $(\mathcal{F}_n)_{n≥1}$-stopping time ?

I need to show that $\left\{\tau \leq n\right\} \in \mathcal{F}_n \; $ for all $n \geq1 $

Answer 1

Let $n \geq1$ $$\{\tau \leq n\}= \bigcup_{k=1}^n{\{Z_k=0\}}$$ where $$Z_k=X_k-Y_k$$

$Z_k$ is $(\mathcal{F}_k)$-measurable, hence your results.