Kunita Watanabe Identity

Question

I am looking for a proof of the following version of Kunita Watanabe Identity:

"Let $M,N \in M_{c,loc}$ and $H$ be a locally bounded previsible process. Then

$[H \cdot M, N ] = H \cdot [M,N]$"

I have read the proof given by Nathanael Berestycki in his Stochastic Chalculus notes, and I am having trouble understanding how he uses the optional stopping theorem. Besides he doesn't show the arguments for going from simple integrands to general $L^2(M)$ processes.

Any reference or help is appreciated.

Answer 1

1. Step 1: Extension from simple (previsible) functions to square integrable function: Let $f \in L^2(M)$, i.e. $$\mathbb{E} \left( \int_0^{\infty} f^2(s) \, d\langle M \rangle(s) \right)< \infty.$$ It follows from the very definition that we can choose simple (previsible) functions $(f_n)_{n \in \mathbb{N}} \subseteq L^2(M)$ such that $f_n \to f$ in $L^2(M)$. Itô's isometry shows that $$\int_0^t f_n(s) \, dM_s \stackrel{L^2(\mathbb{P})}{\to} \int_0^t f(s) \, dM_s$$ for all $t \geq 0$. On the other hand, applying the Kunita-Watanabe inequalities, we get $$\begin{align*} \left|\mathbb{E} \left( \int_0^t (f(s)-f_n(s)) \, d\langle M,N \rangle(s) \right) \right|^2 &\leq \mathbb{E}\int_0^t |f(s)-f_n(s)|^2 \, d\langle M \rangle(s) \cdot \mathbb{E}\int_0^t d\langle N \rangle(s) \\ &\to 0 \quad \text{as} \, n \to \infty. \end{align*}$$ Consequently, we have shown that $$\begin{align*}X^n(t) &:= \left( \int_0^t f_n(s) \, dM_s \right) \cdot N_t - \int_0^t f_n(s) \, d\langle M,N \rangle(s) \\&\stackrel{L^1(\mathbb{P})}{\to} \left( \int_0^t f(s) \, dM_s \right) \cdot N_t - \int_0^t f(s) \, d\langle M,N \rangle(s). \end{align*}$$ Since the left-hand side is for each $n$ is a martingale, we conclude that the right-hand side is a martingale as an $L^1$-limit of martingales. This finishes the proof.
2. Step 2: Extension from square integrable functions to locally square integrable functions (this includes locally bounded functions): Suppose that $f$ is locally square integrable, i.e. there exists a sequence of stopping times $\tau_n$ such that $\tau_n \uparrow \infty$ and $f^{\tau_n} \in L^2(M)$. From the first step, we see that for each fixed $n$ the process $$X^n(t) := \left( \int_0^{t \wedge \tau_n} f(s) \, dM_s \right) \cdot N_{t \wedge \tau_n} - \int_0^{t \wedge \tau_n} f(s) \, d\langle M,N \rangle(s)$$ is a martingale. Now the claim follows from the very definition of the covariation.
3. Step 3: Extension from martingales to local martingales. Without loss of generality, we can pick a sequence of stopping times $(\sigma_n)$ such that the stopped processes $M^{\sigma_n}$ and $N^{\sigma_n}$ are martingales. Noting that $$\int_0^t f_n(s) \, dM_s^{\sigma_n} = \int_0^{t \wedge \sigma_n} f_n(s) \, dM_s$$ it follows as in the second step from the very definition that $[f \bullet M,N] = f \bullet [M,N]$. If you have any further questions regarding this localization procedure, please add more details to your question.