Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2007 / A / 1 / c / Solution

The increment $B_t-B_s$ is independent of ${\mathcal{F}}_s$, with characteristic function $e^{-{\xi }^2(t-s)/2}$. Since $B_s$ is measurable at time $s$,The heat kernel identity in the hint gives $(P_{u}f)(x)=e^{i\xi x}{e}^{-{\xi }^{2}u/2}$, so this is exactly $(P_{t-s}f)(B_s)$. Multiplying by the deterministic factor $e^{{\xi }^{2}t/2}$ givesThe process is adapted, and $|M_t|=e^{{\xi }^{2}t/2}$ is finite at each fixed time. Thus it is a complex martingale, meaning that its real and imaginary parts are integrable martingales.