A <martingale> relative to a <filtration> $(\mathcal F_n)$ is an adapted sequence of integrable random variables with $E[X_{n+1}\mid\mathcal F_n]=X_n$ almost surely for every $n$. Adapted means $X_n$ is $\mathcal F_n$-measurable. Iterating gives $E[X_m\mid\mathcal F_n]=X_n$ for all $m\geq n$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 26K / i / Solution

 Solution (source code)