Consider the stopped process: $X^S_t(\omega):=X_{S\wedge t}(\omega)$.
If $X$ is a supermartingale and $S$ a bounded stopping times, then the stopped process $X^S$ is also a supermartingale.
My attempt:
- By a lemma, since $S\wedge t$ is a stopping time, we know that $X_{S\wedge t}$ is $\mathcal{F}_{S\wedge t}$-measurable.
- $\forall_{m\geq n} \ S\wedge n \leq S\wedge m \Rightarrow E(X_{S\wedge m} \mid \mathcal{F}_{S\wedge n})\leq X_{S\wedge n}$, by the Optional Stopping–Bounded Stopping Times theorem.
- Now, I only need to prove that $X_{S\wedge t}$ is integrable for all $t$... I know that $X_t$ is integrable for all $t$, and that $X_{S\wedge t}=\sum^K_{i=0}X_{i\wedge t}1_{S=i}$, and so, $X_{S\wedge t}$ is integrable.
Is this correct?