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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2018 • Question 6

Question

Math 6010 • Linear Models Fall 2018 • Question 6

Let \(\{X_n:n\geq 1\}\) be a stochastic process adapted to a filtration \(\{\mathcal {F}_n:n\geq 1\}\) and satisfying \(\mathbb {E}[|X_n|]<\infty \) for all \(n\geq 1\). Prove that \(\{X_n:n\geq 1\}\) is a martingale if and only if \(\mathbb {E}[X_T]=\mathbb {E}[X_1]\) for all bounded stopping times \(T\).

Raw LaTeX

Let \(\{X_n:n\geq1\}\) be a stochastic process adapted to a
filtration \(\{\mathcal{F}_n:n\geq1\}\) and satisfying \(\E[|X_n|]<\infty\) for all
\(n\geq1\). Prove that \(\{X_n:n\geq1\}\) is a martingale if and only if
\(\E[X_T]=\E[X_1]\) for all bounded stopping times \(T\).

Topic Tags

Brownian Motion and Stochastic Processes Filtrations and Stopping Times Martingales and Supermartingales Uniform Integrability and Truncation Methods