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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2005 • Question 9

Question

Math 6010 • Linear Models 2005 • Question 9

Suppose \(\{X_n\}_{n=1}^{\infty }\) is a martingale that is bounded in \(L^1(\mathbb {P})\); i.e., \(\sup _n\|X_n\|_1<\infty \). Use Doob’s decomposition to prove that \(X_T\in L^1(\mathbb {P})\) for all stopping times \(T\).

Raw LaTeX

Suppose \(\{X_n\}_{n=1}^{\infty}\) is a martingale that is bounded in
\(L^1(\Prob)\); i.e., \(\sup_n\|X_n\|_1<\infty\). Use Doob's decomposition to prove
that \(X_T\in L^1(\Prob)\) for all stopping times \(T\).

Topic Tags

Exponential Martingales Filtrations and Stopping Times Martingales and Supermartingales Uniform Integrability and Truncation Methods