Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 202 / 5

Let $A=⟨M⟩$ be the quadratic variation and stop at $T_b=\inf \{u\ge 0:A_u>b\}$. Its continuity and monotonicity imply $A_{u\wedge T_b}\le b$ and $A_{T_b}=b$ whenever $T_b<\infty$. The standard martingale isometry after stopping by bounded quadratic variation gives $\mathbb{E}(M_{t\wedge T_b}^2)=\mathbb{E}{A}_{t\wedge T_b}\le b$; in particular the stopped local martingale is square-integrable. This criterion follows by localization and the Doob L2 maximal inequality, which bound the second moment of the stopped supremum by $4b$ and then permit removal of localization.

On $\{A_t\le b\}$ one has $T_b\ge t$, so the original and stopped paths agree through time $t$. Therefore the Doob L2 maximal inequality and the Markov inequality giveThus

Take the stopping timesContinuity gives $|M_{t\wedge S_n}|\le n$, with no overshoot at the first crossing. Stopping preserves the local martingale property, and the bounded case proved in question 4(i) makes each $M^{S_n}$ a continuous true martingale.

Every continuous path is bounded on each compact time interval, so $S_n\uparrow \infty$ almost surely. Hence this is also a localizing sequence consisting of bounded stopped martingales.

For the dyadic power variation of a continuous local martingale, write ${\Delta }_i^{n}M=M_{i{2}^{-n}}-M_{(i-1)2^{-n}}$, ${\delta }_n={\max }_i|{\Delta }_i^{n}M|$ and $Q_n={\sum }_i({\Delta }_i^{n}M{)}^2$. Uniform continuity on $[0,1]$ gives ${\delta }_n\to 0$ almost surely, while the deterministic-partition characterization of quadratic variation gives $Q_n\to ⟨M{⟩}_1$ in probability. In particular $(Q_n)$ is bounded in probability.

If $p>2$, then $Z_n\le {\delta }_n^{p-2}{Q}_n$, and a factor tending to zero in probability times a sequence bounded in probability tends to zero in probability. ThusIf $1<p<2$, the reverse comparison is $Q_n\le {\delta }_n^{2-p}{Z}_n$. The assumed finite $\mathrm{limsup}{Z}_n$ makes $Z_n$ eventually bounded almost surely, so $Q_n\to 0$ almost surely. By uniqueness of a limit in probability, $⟨M{⟩}_1=0$ almost surely. Its monotonicity gives $⟨M{⟩}_t=0$ for $t\le 1$.

To finish without assuming any integrability of $M$, apply the bound from part (i) to both $M$ and $-M$, with $t=1$, and then let $b\downarrow 0$. For each $a>0$ the probability of a positive or negative excursion beyond $a$ is zero. Taking countably many $a\downarrow 0$ proves $M$ is indistinguishable from zero on $[0,1]$. The same argument in fact works for every $0<p<2$ under the stated pathwise boundedness assumption.

The usual quadratic variation convention includes $⟨M{⟩}_0=0$. If $A,C$ are two continuous finite-variation processes with $A_0=C_0=0$ such that $M^2-A$ and $M^2-C$ are continuous local martingales, then $A-C$ is a continuous local martingale of finite variation. It must be constant, and its initial value is zero, so $A=C$ up to indistinguishability of stochastic processes.

Here is a proof of the fact used. Subtract the initial value of a continuous local martingale $N$ of finite variation, and stop so that both $|N|$ and its total-variation process are bounded by deterministic constants. The stopped local martingale is a bounded true martingale, so orthogonality of its increments gives, on any deterministic partition of an interval of $[0,t]$,Continuity and the dominated convergence theorem justify the last limit. Remove stopping and use continuity on a countable dense set of times to conclude $N\equiv 0$.

As literally printed, uniqueness is false without normalization. If $A$ works then $A+c$ also works for any deterministic constant $c$, since subtracting a constant preserves the local martingale property; this is a genuine omitted convention.

With normalization, existence always holds for a continuous local martingale. Its quadratic variation $[M]$, obtained as the limit in probability of squared increments, is continuous, increasing, starts at zero and satisfies $M^2-[M]$ is a continuous local martingale, so taking $⟨M⟩=[M]$ supplies the required finite-variation process.