For a partition $\pi$ of $[0,t]$, apply the first inequality to each increment of the covariation and then the <Cauchy-Schwarz inequality>:
$$
\sum_{\pi}|\Delta[M,N]|
\leq\sum_\pi\sqrt{\Delta[M]\,\Delta[N]}
\leq\sqrt{[M]_t[N]_t}.
$$
Taking the supremum over partitions proves the <Kunita-Watanabe inequality>
$$
V_t([M,N])\leq\sqrt{[M]_t[N]_t}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 5 / a / 2 / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-202/5/a/2/solution