Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 202 / 1 / f / Solution

Each $A^{ϵ}$ is continuous, adapted, and increasing. From ucp convergence one can choose a subsequence that converges uniformly almost surely on every compact interval. Its limit $A$ is therefore also continuous and increasing, hence a finite-variation process. The stochastic integral $M_t={\int }_0^t\mathrm{sgn}(B_s)d{B}_s$ is a continuous local martingale, and part e gives the semimartingale decompositionConsequently $|B|$ is a semimartingale. In fact, comparison with the Tanaka formula identifies $A_t$ as the local time of a semimartingale $L_t^0(B)$.