Put $M_n=\mathbb E[Z\mid\mathcal F_n]$. Each $M_n$ is $\mathcal F_n$-measurable and integrable, with $|M_n|\leq\mathbb E[|Z|\mid\mathcal F_n]\leq1$. Since a <filtration> is increasing, the <tower property of conditional expectation> gives
$$
\mathbb E[M_{n+1}\mid\mathcal F_n]
=\mathbb E[\mathbb E[Z\mid\mathcal F_{n+1}]\mid\mathcal F_n]
=\mathbb E[Z\mid\mathcal F_n]=M_n.
$$
Thus $(M_n)$ is the <conditional-expectation martingale> associated with $Z$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 201 / 1 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-201/1/a/solution