Prove $Y_S$ is integrable if $Y$ is a supermartingale and $S$ is a bounded stopping time.

Question

Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}_{n \in \mathbb{N}}, \mathbb{P})$,

let $Y = ({Y_n})_{n \in \mathbb{N}}$ be a/an $(\{\mathscr{F_n}\}_{n \in \mathbb{N}}, \mathbb{P})$-supermartingale.

Prove $Y_S$ is integrable if $Y$ is a supermartingale and $S$ is a bounded stopping time.

So far all I was able to show is that $Y_{S \wedge n}$ is integrable and $E[Y_S] \le E[Y_0]$.

Hints pls?

Answer 1

Hints:

1. Since $S$ is a bounded stopping time, there exists $N \in \mathbb{N}$ such that $S(\omega) \leq N$ for all $\omega \in \Omega$. Show that $$\mathbb{E}(|Y_S|) = \sum_{k=0}^n \mathbb{E}(|Y_k| 1_{\{S=k\}}).$$
2. Conclude $$\mathbb{E}(|Y_S|) \leq \sum_{k=0}^N \mathbb{E}(|Y_k|) \leq (N+1) \max_{k=0,\ldots,N} \mathbb{E}(|Y_k|)<\infty.$$

Note that this holds for any stochastic process $(Y_n)_{n \in \mathbb{N}}$ (we do not need the supermartingale property).