Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 202 / 2

Let $(X_n)$ be Cauchy in the normChoose a subsequence $(X_{n_k})$ for whichTonelli theorem impliesalmost surely. The subsequence therefore converges uniformly on $[0,\infty )$, outside one null event, to a continuous process $X$. For each $t$, $X_t$ is the almost-sure limit of ${\mathcal{F}}_t$-measurable variables; completeness of the filtration makes the chosen version adapted. The same summable bound and monotone convergence theorem show that $\mathbb{E}{\sup }_t|X_t|<\infty$ and that $X_{n_k}\to X$ in norm.

Since the original sequence is Cauchy, the usual triangle argument upgrades convergence of the subsequence to $X_n\to X$ in norm. Thus the space of indistinguishability classes is a Banach space.

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Fix a deterministic horizon $T$ and $\epsilon >0$. For every localization index $N$,by Markov inequality. Because $T_N\uparrow \infty$ almost surely, the first term tends to zero as $N\to \infty$. For fixed $N$, the second tends to zero as $n\to \infty$. Taking first the limit superior in $n$ and then $N\to \infty$ provesin probability. This is precisely uniform convergence on compacts in probability.

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Use the continuous semimartingale decomposition $X=X_0+M+A$, where $M$ is a continuous local martingale and $A$ is a continuous adapted finite-variation process. Pointwise limits preserve predictability, so $H$ is predictable; it is bounded by the common bound for the $H_n$.

Localize so that $[M{]}_{\infty }$ and the total variation $V(A{)}_{\infty }$ are bounded. The Doob L2 maximal inequality and the Itô isometry giveby the dominated convergence theorem. For the finite-variation part,almost surely, again by dominated convergence, now for each sample path. Hence the two integrals converge uniformly in probability after every localization. Part (b) removes the localization and provesu.c.p.

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PutThen $f_n\to |\cdot |$ uniformly, $f_n^{\prime }(x)=\tanh (nx)$, and $f_n^{\prime \prime }(x)=n{\mathrm{sech}}^2(nx)$. Itô formula givesThe bounded predictable integrands $\tanh (nX)$ converge pointwise to $\mathrm{sgn}(X)$, with $\mathrm{sgn}(0)=0$. Part (c) therefore makes the stochastic integrals converge u.c.p. The left-hand side converges u.c.p. to $|X|$, so the increasing continuous processesalso converge u.c.p. Their limit $A$ has a continuous increasing version: extract almost-sure locally uniform convergence from each compact interval and use a diagonal argument. We obtainthe Tanaka formula with $A=L_t^0(X)$. It expresses $|X|$ as a continuous local martingale plus a continuous finite-variation process, so $|X|$ is a continuous semimartingale.

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