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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2017 • Question 9

Question

Math 6010 • Linear Models Fall 2017 • Question 9

Let \(M_n\) be an \(L^1\)-bounded martingale with respect to a filtration \(\mathcal {F}_n\). Prove that there exist two non-negative martingales \(Y_n\) and \(Z_n\) (in the same filtration) such that \(M_n=Y_n-Z_n\). Hint: Use \(Z_{n,j}=\mathbb {E}[|M_j|\mid \mathcal {F}_n]\) for \(j\geq n\).

Raw LaTeX

Let \(M_n\) be an \(L^1\)-bounded martingale with respect to a
filtration \(\mathcal{F}_n\). Prove that there exist two non-negative martingales
\(Y_n\) and \(Z_n\) (in the same filtration) such that \(M_n=Y_n-Z_n\). Hint: Use
\(Z_{n,j}=\E[|M_j|\mid\mathcal{F}_n]\) for \(j\geq n\).

Topic Tags

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