I was succesful at showing that the quadratic covariation $\langle\cdot ,\cdot \rangle_t$ is a positiv semidefinit, symmetric and bilinear form for each $t$ on the set of local martigales. So the Cauchy-Schwarz inequality gives me: \begin{align*} |\langle M,N\rangle_t| \leq \sqrt{\langle M,M\rangle_t}\sqrt{\langle N,N\rangle_t}=\sqrt{\langle M\rangle_t}\sqrt{\langle N\rangle_t} \end{align*} That is already very nice, but for the proof of Kunita-Watanabe inequality my Professor uses following:
\begin{align*} |\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} \end{align*} Can someone explain me why this is true? Thanks!
Hint: Fix $s \leq t$. Show that $$\langle M,N \rangle_t^s := \langle M,N \rangle_t - \langle M,N \rangle_s $$ defines a symmetric positiv definite bilinear form.
Alternatively: Apply the first inequality (the one you already proved) to the shifted processes $$\tilde{M}_t := M_{t+s}-M_s \qquad \quad \tilde{N}_t := N_{t+s}-M_s, \qquad t \geq 0.$$