Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 202 / 5 / 1 / Solution

A simple predictable process has the formwhere each bounded ${\xi }_i$ is ${\mathcal{F}}_{t_i}$-measurable. DefineIndependent centered Brownian increments show directly by conditioning that this is a martingale. The same conditional expansion, using $\mathbb{E}[(B_v-B_u{)}^2\mid {\mathcal{F}}_u]=v-u$, shows that