# Predictable process with stopping time index

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?

• What have you tried so far? Why do you expect this to be true? Commented May 18, 2023 at 16:20
• @user6247850 have edited Commented May 18, 2023 at 16:28

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.