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

For every real $\lambda$, positivity of quadratic variation givesThe discriminant of this quadratic is nonpositive, soContinuity lets the almost-sure assertion hold simultaneously for every $t$.

For a partition $\pi$ of $[0,t]$, apply the first inequality to each increment of the covariation and then the Cauchy-Schwarz inequality:Taking the supremum over partitions proves the Kunita-Watanabe inequality