 Solution

A process $M_n$ is a $\mathbb{P}$-martingale relative to the filtration $({\mathcal{F}}_n)$ when it is adapted, each $M_n$ is integrable, andHere adapted means $M_n$ is ${\mathcal{F}}_n$-measurable, and integrable means ${\mathbb{E}}_{\mathbb{P}}|M_n|<\infty$. Thus its present value is the conditional mean of its next value given all present information.