Let $(A_n)$ be a predictable process and $\tau$ a stopping time. Is it true that $(A_{n\land\tau})$ is also predictable? How would I argue this? Thanks.
My attempt: I think $A_{n\land\tau}\in\{A_1,...,A_n\}$, and each of them is $F_{n-1}$ - measurable. Hence the statement is true. Would this be correct?
The range being $F_{n-1}$ measurable isn't quite enough because the specific value it takes will depend on $\tau$. However, we have $A_{n \wedge \tau} = A_{\tau} 1_{\tau \le n-1} + A_n 1_{\tau > n-1}$. The event $\{\tau \le n-1\}$ is $F_{n-1}$ measurable, and $A_\tau$ is $F_{n-1}$ measurable on $\{\tau \le n-1\}$, so the first term is $F_{n-1}$ measurable. The second term is also measurable with similar reasoning, and using that $A$ is predictable to conclude $A_n$ is $F_{n-1}$ measurable.