Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 202 / 2 / a / Solution

Use polarization identity of the quadratic variations:This is continuous and adapted, starts at zero, and has finite variation on every compact interval because it is a difference of two nondecreasing processes. The defining quadratic variation property gives that $(M+N{)}^2-[M+N]$ and $(M-N{)}^2-[M-N]$ are continuous local martingales; their difference divided by four is $MN-[M,N]$. This proves existence without assuming a product formula involving an as yet undefined quadratic covariation.

If $C$ and $D$ both satisfy the requirements, $C-D$ is a continuous finite-variation process and a local martingale. By continuous finite-variation local martingale is constant, it is constant; its initial value is zero, so $C=D$ up to indistinguishability of stochastic processes.

There is a minor initial-value convention in this characterization. The displayed uncentred local martingale identity is directly valid when $M_0=N_0=0$, or when the initial terms have the requisite integrability. With arbitrary finite initial values, centre the quadratic variation construction and use $MN-M_0{N}_0-[M,N]$, which starts at zero. Adding back $M_0{N}_0$ requires it to be integrable under the usual definition of a local martingale; the quadratic covariation itself is unaffected by centring.