How to show that a stochastic process is predictable?

Question

Suppose $(X_n, \mathbb{F}_n)$ is a submartingale and $\tau$ is a stopping time. Now look at the process $H_n$ given by: $$H_n= \left\{ \begin{array}{ll} 1 & \text{if }n\leq\tau \\ 0 & \text{if }n>\tau \\ \end{array} \right. $$ How can I show that $H_n$ is predictable? Intuitively it makes really good sense. I tried working out how to show that $H_n$ is either $\mathbb{F}_{n-1}$- or $\mathbb{F_{\tau}}$-measurable, but got nowhere. In addition a hint is given: what is the relation between $(H\cdot X)_n$ and $X_{\tau\wedge n}$.

Answer 1

I think that hint massively over-complicates it; we can just show $H_n$ is $\mathcal F_{n-1}$ measurable directly. Notice that

\begin{align*} \{H_n = 0 \} &= \{ \tau < n \} = \{\tau \le n-1\} \in \mathcal F_{n-1} \end{align*}

by the definition of a stopping time. Since $H_n$ can only take on two distinct values, this proves $H_n$ is $\mathcal F_{n-1}$ measurable.