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

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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