For every real $\lambda$, positivity of quadratic variation gives
$$
0\leq[M+\lambda N]_t
=[M]_t+2\lambda[M,N]_t+\lambda^2[N]_t.
$$
The discriminant of this quadratic is nonpositive, so
$$
|[M,N]_t|\leq\sqrt{[M]_t[N]_t}.
$$
Continuity lets the almost-sure assertion hold simultaneously for every $t$.


Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 5 / a / 1 / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-202/5/a/1/solution