Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 36 / 1

Take a complete, right-continuous filtration $({\mathcal{F}}_t)$, an increasing family of sigma-algebras. Adapted means that $X_t$ is ${\mathcal{F}}_t$-measurable at each $t$; càdlàg means right-continuous with left limits. A local martingale is an adapted càdlàg process $X$, with integrable initial value, for which there are stopping times ${\tau }_n\uparrow \infty$ almost surely such that each stopped process $X_t^{{\tau }_n}=X_{t\wedge {\tau }_n}$ is a true martingale. Such a sequence is a localizing sequence. A true martingale is integrable at each time and satisfies $\mathbb{E}(X_t\mid {\mathcal{F}}_s)=X_s$ for $s\le t$.

The necessary and sufficient condition is that $X$ is a Class DL process:A family $\mathcal{Z}$ is uniformly integrable if its members are integrable and ${\sup }_{Z\in \mathcal{Z}}\mathbb{E}(|Z|1_{\{|Z|>K\}})\to 0$ as $K\to \infty$. A stopping time $\tau$ satisfies $\{\tau \le t\}\in {\mathcal{F}}_t$ for all $t$.

For necessity, on $[0,T]$ the optional stopping theorem gives $X_{\tau }=\mathbb{E}(X_T\mid {\mathcal{F}}_{\tau })$; conditional expectations of one integrable variable form a uniformly integrable family. For sufficiency, $X_{t\wedge {\tau }_n}\to X_t$ almost surely, and Class DL upgrades this convergence to $L^1$ on each finite horizon. Pass to the limit in $\mathbb{E}(X_{t\wedge {\tau }_n}\mid {\mathcal{F}}_s)=X_{s\wedge {\tau }_n}$ to obtain the martingale identity. Requiring uniform integrability of all times on the entire half-line would be stronger than necessary; Brownian motion is already a counterexample to that stronger requirement.

The Lévy characterization of multidimensional Brownian motion says that a continuous adapted ${\mathbb{R}}^d$-valued process $Z$, with $Z_0=0$, is standard Brownian motion relative to the given filtration if and only if its coordinates are local martingales andFor Brownian motion, independent centered Gaussian increments give the martingale property of each coordinate and of $Z_t^i{Z}_t^j-{\delta }_{ij}t$. The defining uniqueness of quadratic covariation consequently gives the bracket condition.

Conversely, suppose the local martingale and bracket conditions hold. For $\theta \in {\mathbb{R}}^d$, the Itô formula givessince the bracket term cancels the time derivative. On $[0,T]$, $|E_t|\le e^{|\theta {|}^{2}T/2}$, so its real and imaginary parts are bounded local martingales and hence true martingales. Thus, for $s\le t$,This is the characteristic function of $N(0,(t-s)I_d)$ and is independent of ${\mathcal{F}}_s$. Uniqueness of characteristic functions shows that $Z_t-Z_s$ has that Gaussian distribution and is independent of the past. Iterating over ordered times gives independent Gaussian increments; together with continuous paths and $Z_0=0$, these are precisely the defining properties of standard Brownian motion.

First justify that the logarithm is defined at all times from one onward. For planar Brownian motion started at $z\ne 0$, stop on exiting the annulus $\epsilon <|x|<R$, where $\epsilon <|z|<R$. This exit time $\rho$ is finite almost surely: the stopped identity $|B_t{|}^2-2t$ gives $\mathbb{E}(t\wedge \rho )\le R^2/2$. The function $f(x)=\log |x|$ is harmonic on the annulus, so its stopped process is a bounded martingale. Optional stopping givesHitting zero before the outer circle would require hitting every inner circle first. It therefore has probability zero. Taking a countable sequence of outer radii tending to infinity proves that planar Brownian motion avoids a fixed point. Since $B_1$ has a density and is nonzero almost surely, condition on ${\mathcal{F}}_1$ to apply this result to the future path.

For $x\ne 0$, $\nabla f(x)=x/|x{|}^2$ and $\Delta f(x)=0$ in two dimensions. The Itô formula on annuli therefore givesTo localize explicitly, let ${\tau }_n=\inf \{t\ge 1:|B_t|\notin (1/n,n)\}$. These stopping times increase to infinity almost surely: on each compact time interval the continuous path has a finite maximum and, by point avoidance, a positive minimum radius. Up to ${\tau }_n$, the stochastic integrand has norm at most $n$. Its stopped integral is square-integrable on every finite horizon. The initial value $\log |B_1|$ is integrable, as follows from the Rayleigh-density calculation in part d. Hence $X^{{\tau }_n}$ is a true martingale from time one and $X$ is a local martingale. This also gives its quadratic variation from time one as ${\int }_1^t|B_s{|}^{-2}ds$.

Brownian scaling gives $|B_t|\stackrel{d}{=}\sqrt{t}R$, where $R=|B_1|$ has Rayleigh distribution with density $r{e}^{-r^2/2}$, $r>0$. For every $p>0$,Near zero this is bounded by ${\int }_0^1|\log r{|}^{p}rdr={\int }_0^{\infty }{u}^p{e}^{-2u}du$, which is finite; near infinity the Gaussian tail dominates any logarithmic power. The inequality $|a+b{|}^p\le C_p(|a{|}^p+|b{|}^p)$, valid also for $0<p<1$, now yieldsLet $c=\mathbb{E}\log R$, a finite constant. The same scaling identity givesAn integrable martingale has constant expectation, so the logarithm of planar Brownian radius is a strict local martingale on $[1,\infty )$. Fixed-time integrability, even of every positive power, does not imply the stopping-time uniform integrability required in part a.