For every real $\lambda$, the <quadratic variation> of $\lambda(M-M_s)+(N-N_s)$ on $[s,t]$ is nonnegative:
$$
\lambda^2\Delta\langle M\rangle
+2\lambda\Delta\langle M,N\rangle
+\Delta\langle N\rangle\geq0.
$$
Its discriminant is therefore nonpositive. This gives the pathwise <Kunita-Watanabe inequality>
$$
\boxed{|\langle M,N\rangle_t-\langle M,N\rangle_s|
\leq\sqrt{\langle M\rangle_t-\langle M\rangle_s}
\sqrt{\langle N\rangle_t-\langle N\rangle_s}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 202 / 4 / 1 / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-202/4/1/solution