# Notation of the Kunita–Watanabe inequality is not clear

I have confusion on understanding the Kunita–Watanabe inequality and the notation used there.

In my script we defined the covariation process $\langle M,N\rangle$ and it then says:

The Kunita-Watanabe inequality shows that the variation $|\langle M,N\rangle|$ of $\langle M,N\rangle$ is $P$-a.s. absolutely continuous with respect to both $\langle M\rangle$, $\langle N\rangle$ .

1. So are we speaking of the variation of the covariation process? Is this defined as: $$|\langle M,N\rangle|=\lim_{\Vert P\Vert\rightarrow 0}\sum_{k=1}^n(\langle M,N\rangle_{t_k}-\langle M,N\rangle_{t_{k-1}})$$

It confuses me since these extra $|\quad|$ are used in the Kunitaba-Watanabe inequality in my script but for instance on Wikipedia they are missing in the left hand side integrator. We defined the KWT as:

Let M, N be continuous local martingales and H,K measurable processes. Then $$\int_0^t | H_s | | K_s || \,\mathrm{d} \langle M,N \rangle_s |\leq \sqrt{\int_0^t H_s^2 \,\mathrm{d} \langle M \rangle_s} \sqrt{\int_0^t K_s^2 \,\mathrm{d} \langle N \rangle_s}$$

1. The $| H_s |$ means the absolute values of $H_s$ or are we using here the variation again?

Thanks for clarification.

• 2. Yes. (The $|\cdot|$ is being used in two senses in the inequality: i) Absolute value when applied to a process like $H$; ii) toal variation when applied to a signed measure like $d\langle M,N\rangle$.) – John Dawkins Aug 5 '17 at 17:08
• @JohnDawkins Thank you very much. – Matriz Aug 5 '17 at 19:05