Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 201 / 3

Let $(M_k,{\mathcal{F}}_k{)}_{k=0}^N$ be a martingale, let $a<b$, and let $U_N[a,b]$ count the completed upcrossings of $[a,b]$ by time $N$. Doob upcrossing inequality statesEquivalent conventions for a process with a prescribed initial holding add the corresponding endpoint term.

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Assume ${\sup }_n\mathbb{E}|M_n|<\infty$. For every pair of rationals $a<b$, Doob upcrossing inequality givesby monotone convergence theorem. Thus every rational interval is upcrossed only finitely often almost surely. If a real sequence has distinct limit inferior and limit superior, it completes infinitely many upcrossings of some rational interval between them. Hence $M_n$ converges in the extended real line almost surely.

Fatou lemma givesso the limit is finite almost surely and integrable. This is the $L^1$-bounded form of the Martingale convergence theorem.

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Let $(M_t{)}_{t\ge 0}$ be a right-continuous continuous-time martingale with ${\sup }_t\mathbb{E}|M_t|<\infty$. Restricting it to the nonnegative rational times gives a countable martingale. The argument of part (b), applied on successively finer rational grids, shows that it has a finite almost-sure limit as the rational time tends to infinity. Right-continuity and the upcrossing characterization prevent the values at arbitrary times from having a different limit. Thus $M_t$ converges almost surely to a finite integrable random variable as $t\to \infty$, which is the Continuous-time martingale convergence theorem.

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Let $(B_t)$ be Brownian motion in ${\mathbb{R}}^3$ started at a nonzero point. The function $h(x)=|x{|}^{-1}$ is a positive harmonic function on ${\mathbb{R}}^3\setminus \{0\}$ because its Laplacian vanishes there. Stopping on annuli and applying Itô formula shows that $h(B_t)$ is a local martingale. Letting the inner boundary shrink to zero also shows that three-dimensional Brownian motion does not hit the origin.

A positive local martingale is a supermartingale, so $h(B_t)$ is $L^1$-bounded by $h(B_0)$. The upcrossing proof from part (c), which applies verbatim to a positive supermartingale, therefore gives a finite almost-sure limit

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By Brownian scaling,where $Z$ has the standard three-dimensional multivariate normal distribution. The right-hand side tends to zero in probability, since $Z$ has no atom at the origin. Part (d) gives almost-sure convergence to $Y$, which also implies convergence in probability to $Y$. Uniqueness of a limit in probability therefore gives $Y=0$ almost surely. Hence $|B_t|\to \infty$ almost surely, proving the transience of Brownian motion in dimension at least three in dimension three.

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