Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 30 / 3 / b

Fix $s<t$ and an event $A\in {\mathcal{F}}_s$. Multiplying the conditional characteristic function identity by $1_A$ and taking expectation givesThe left side is the Fourier transform of the finite measureBy the uniqueness theorem for characteristic functions, this measure equals $\mathbb{P}(A)$ times the $N(0,t-s)$ distribution. Thus, for every Borel set $D$,Taking $A$ to be the whole sample space identifies the increment's law, and the full identity proves independence from ${\mathcal{F}}_s$. Together with the assumed adaptation, continuity, and initial value, these are exactly the defining Brownian properties. This proves the conditional characteristic-function criterion for Brownian increments.