Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2018 • Question 7

Question

Math 6010 • Linear Models Fall 2018 • Question 7

Let \(\{X_n:n\geq 1\}\) be a martingale with respect to filtration \(\{\mathcal {F}_n:n\geq 1\}\), and let \(T\) be a stopping time with respect to \(\{\mathcal {F}_n:n\geq 1\}\). Prove the following two forms of the optional stopping theorem:

  1. If \(T\) is almost surely bounded then \(\mathbb {E}[X_T]=\mathbb {E}[X_1]\),
  2. If \(\mathbb {E}[\max _{1\leq n\leq T}|X_n|]<\infty \), then \(\mathbb {E}[X_T]=\mathbb {E}[X_1]\).

Raw LaTeX

Let \(\{X_n:n\geq1\}\) be a martingale with respect to filtration
\(\{\mathcal{F}_n:n\geq1\}\), and let \(T\) be a stopping time with respect to
\(\{\mathcal{F}_n:n\geq1\}\). Prove the following two forms of the optional
stopping theorem:
\begin{enumerate}
\item[a)] If \(T\) is almost surely bounded then \(\E[X_T]=\E[X_1]\),
\item[b)] If \(\E[\max_{1\leq n\leq T}|X_n|]<\infty\), then
\(\E[X_T]=\E[X_1]\).
\end{enumerate}

Topic Tags

Convergence in Probability and Almost Sure Convergence Filtrations and Stopping Times Martingales and Supermartingales Uniform Integrability and Truncation Methods