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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2008 • Question 4

Question

Math 6010 • Linear Models Spring 2008 • Question 4

Let \(X_1,X_2,\ldots \) be a sequence of integrable random variables defined on the same probability space. Show that \(\{X_n\}_{n\geq 1}\) is a submartingale if and only if \(\mathbb {E}[X_T]\geq \mathbb {E}[X_1]\) for all bounded stopping times \(T\). (Take the filtration to be the natural one, \(\mathcal {F}_n=\sigma (X_1,\ldots ,X_n)\).)

Raw LaTeX

Let \(X_1,X_2,\ldots\) be a sequence of integrable random variables
defined on the same probability space. Show that \(\{X_n\}_{n\geq1}\) is a
submartingale if and only if \(\E[X_T]\geq\E[X_1]\) for all bounded stopping
times \(T\). (Take the filtration to be the natural one,
\(\mathcal{F}_n=\sigma(X_1,\ldots,X_n)\).)

Topic Tags

Filtrations and Stopping Times Martingales and Supermartingales Measure-Theoretic Probability Uniform Integrability and Truncation Methods