Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 38 / 4 / c / Solution

A simple example isfor a standard Brownian motion $B$. The Itô formula gives $M_t=2{\int }_0^t{B}_{s}d{B}_s$, a continuous square-integrable martingale on every finite horizon. For $h>0$, write $\Delta =B_{s+h}-B_s$, independent of the past and normal with variance $h$. ThenSince $B_s^2=M_s+s$, this conditional second moment is already a nonconstant function of the martingale's own past for $s>0$. The tower property of conditional expectation gives the same expression on conditioning on that own past. Were the increments independent, that conditional second moment would be constant. Thus the increments are not independent.