The <Kunita-Watanabe inequality> applied to the <continuous local martingales> $M,N$ gives directly
$$
V_t([M,N])\leq[M]_t^{1/2}[N]_t^{1/2}.
$$
Equivalently, with the clock and densities from part (i), positivity of the covariance matrix gives $|c|\leq\sqrt{ab}$, and the <Cauchy-Schwarz inequality> gives
$$
V_t=\int_0^t|c|dC\leq\int_0^t\sqrt{ab}\,dC\leq\left(\int_0^ta\,dC\right)^{1/2}\left(\int_0^tb\,dC\right)^{1/2}=[M]_t^{1/2}[N]_t^{1/2}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 202 / 2 / b / iii / Solution

{c}

For continuous local martingales, the <total-variation process> of their <quadratic covariation> satisfies
$$
V_t([M,N])\leq[M]_t^{1/2}[N]_t^{1/2}.
$$
It is the <Cauchy-Schwarz inequality> for the matrix-valued measure formed by their <quadratic variations> and <quadratic covariation>.

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