Concerning the optional sampling theorem out of Karatzas, Shreve BMSC:
I need help understanding why $(X_{S_n}, \mathfrak{F}_{S_n})_n$ is a backward submartingale. I could show that it is $(\mathfrak{F}_{S_n})_n$ adapted. It is left to show that every $X_{S_n}$ is integrable and that the backward submartingale property $\mathbb{E}[X_{S_n}|\mathfrak{F}_{S_{n+1}}]\geq X_{S_{n+1}}$ satisfied.
Questions:
- How do I show that $X_{S_n}$ is integrable?
- Is my approach to show the backward submartingale property valid? If yes, where can I find a proof of the discrete optional sampling theorem for unbounded optional times (other than Chung)?
Approaches:
I know that $\infty<\mathbb{E}[X_0]\leq\mathbb{E}[X_{S_n}]\leq\mathbb{E}[X_\infty]<\infty$. But now I need to have an estimate for $\mathbb{E}[|X_{S_n}|]$, but the observation beforehand goes in the wrong direction. I tried to partition $$\mathbb{E}[|X_{S_n}|]=\mathbb{E}[\sum_{k=1}^\infty{|X_{S_n}|\chi_{\{\frac{k-1}{2^n}\leq S<\frac{k}{2^n}\}}+|X_{S_n}|\chi_{\{S=\infty\}}}]$$ but this leads to the questions of what are $\mathbb{P}[\frac{k-1}{2^n}\leq S<\frac{k}{2^n}]$ and $\mathbb{P}[S=\infty]$. I think this does not lead into the right direction.
By the optional sampling theorem in the discrete version we know that $X_{S_{n+1}}\leq \mathbb{E}[X_{S_n}|\mathfrak{F}_{S_{n+1}}]$ almost surely as $S_{n+1}\leq S_n$.