Optional Sampling Theorem in Continuous Time

Question

This is the buildup to establishing the Optional Sampling Theorem for continuous time. Let $\mathbb{F}$ denote a filtration and $(X_t)_{t\geq 0}$ be a right continuous supermartingale. Let $\sigma$ and $\tau$ be bounded stopping times, and define $\sigma_n=\frac{\lceil 2^{n}\sigma\rceil}{2^n}$, (likewise for $\tau_n$). We need to show that $\mathbb{E}[X_{\tau_n}|\mathcal{F}_{\sigma_m}]\to\mathbb{E}[X_{\tau_n}|\mathcal{F}_{\sigma}]$ a.s. as well as in $L^1$. So far, the only thing that is clear to me is that we should have $X_{\sigma_m}\to X_{\sigma}$ a.s throgh right continuity. Not quite sure how to proceed

Answer 1

The statement is a consquence of Lévy's backwards convergence theorem

Let $Y \in L^1$ be an integrable random variable and $\mathcal{G}_0 \supseteq \mathcal{G}_1 \supseteq \dots$ a decreasing sequence of $\sigma$-algebras. If we set $\mathcal{G}_{\infty} := \bigcap_{j \geq 1} \mathcal{G}_j$, then $$\mathbb{E}(Y \mid \mathcal{G}_j) \xrightarrow[]{j \to \infty} \mathbb{E}(Y \mid G_{\infty})$$ almost surely and in $L^1$.

In your setting, $\mathcal{G}_j := \mathcal{F}_{\sigma_j}$ and $Y=X_{\tau_n}$ for fixed $n$. Check that $(\mathcal{G}_j)_{j \geq 1}$ is a decreasing sequence of $\sigma$-algebras (here you have to use that $\sigma_{n+1} \leq \sigma_n$) and that $$\mathcal{F}_{\sigma} = \bigcap_{j \geq 1} \mathcal{F}_{\sigma_j}$$ (for this you have to use that $\sigma = \inf_j \sigma_j$). Applying Lévy's backward convergence theorem then proves the assertion.