Math 6040 • Probability

Let \(\{\mathcal{F}_{n}\}_{n \ge 0}\) be a filtration on the probability space \((\Omega, \mathcal{F}, P)\), let \(X \in L^{1}(\Omega, \mathcal{F}, P)\),
and define \(M_{n} = \mathbb{E}[X \mid \mathcal{F}_{n}]\). Let \(\tau\) be a stopping time and recall the definition
\[
\mathcal{F}_{\tau} =
\{A \in \mathcal{F} : A \cap \{\tau = n\} \in \mathcal{F}_{n}\ \forall n \ge 0\}.
\]

\begin{enumerate}

\item
Show that \(1_{\{\tau < \infty\}}M_{\tau}\) is integrable.


\item
Show that \(\mathbb{E}[X \mid \mathcal{F}_{\tau}] = 1_{\{\tau < \infty\}}M_{\tau} + 1_{\{\tau = \infty\}}X\). Do not forget the necessary
measurability condition.

\end{enumerate}