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

The raw natural filtration isEquivalently it is the smallest sigma-algebra making every observation $B_u$, $u\le t$, measurable. As $s\le t$ enlarges the set of observations, ${\mathcal{F}}_s^B\subseteq {\mathcal{F}}_t^B$. It is the smallest filtration to which the process is adapted. If the usual conditions for a filtration are imposed, take the completion by the probability-zero sets and its right-continuous augmentation. Brownian independent increments remain valid for this usual augmentation. Adding arbitrary future information is not part of the generated filtration.

Independent increments give $B_s\sim N(0,s)$ independent of $B_t-B_s\sim N(0,t-s)$. Their joint probability density function at $(x,v)$ is the productThe change of variables $(x,v)\mapsto (x,y)=(x,x+v)$ has Jacobian determinant one. Therefore the joint probability density function of $(B_s,B_t)$ isIn particular these coordinates are correlated, with covariance $s$, although the first pair has independent coordinates.

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.