Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2007 / A / 2

Apply Itô's lemma to $F(t,x)=e^{i\xi x+{\xi }^{2}t/2}$. Since $F_t={\xi }^{2}F/2$, $F_x=i\xi F$ and $F_{xx}=-{\xi }^{2}F$, the drift terms cancel:This is initially a local martingale. On every finite interval $[0,T]$,so the stochastic integral is a true square-integrable martingale. Conditional expectation gives $\mathbb{E}(M_t\mid {\mathcal{F}}_s)=M_s$; divide by the nonzero ${\mathcal{F}}_s$-measurable factor $e^{i\xi B_s+{\xi }^{2}t/2}$ to obtain

If $X$ is Brownian motion with respect to the given filtration, independent Gaussian increments give the identity immediately.

Conversely, fix $s<t$ and let $\Delta =X_t-X_s$. Taking expectations gives its characteristic function $e^{-{\xi }^2(t-s)/2}$, so $\Delta \sim N(0,t-s)$. To prove independence, for any $A\in {\mathcal{F}}_s$ define the finite measure ${\nu }_A(E)=\mathbb{P}(A\cap \{\Delta \in E\})$. The conditional identity givesUniqueness of Fourier transforms of finite measures implies ${\nu }_A=\mathbb{P}(A)N(0,t-s)$. This is independence of $\Delta$ from ${\mathcal{F}}_s$. Iterating over times proves independent increments. Together with the stated continuity and $X_0=0$, this gives standard Brownian motion.

Here a process with the stated filtration is understood to be adapted, as required when saying it is Brownian with respect to that filtration. If adaptedness is not implicit, it must be added: an ordinary Brownian motion with a trivial filtration satisfies the displayed conditional characteristic function but is not adapted to that filtration.

Lévy characterization of Brownian motion says that a continuous local martingale $M$ with $M_0=0$ is standard Brownian motion with respect to its filtration if its quadratic variation is $⟨M{⟩}_t=t$. Equivalently one may require $M_t^2-t$ to be a local martingale.

Any weak solution has the integral form $Z_t={\int }_0^t\mathrm{sgn}(Z_s-2)d{W}_s$. Its bounded integrand makes it a continuous local martingale starting at zero, andThe specified value of the sign at zero has square one as well. Lévy characterization of Brownian motion therefore proves that every such weak solution is Brownian motion. This is a statement about its law and filtration; it does not assert $Z=W$ or provide a strong solution as a function of a specified driver.