I have a question about generalization of the Kunita Watanabe decomposition. I've learned the following version:
Let $M$ be a continuous local martingale. Then every continuous local martingale $N$ can be uniquely written as $$ N=H\bullet M + L$$ where $H \bullet M:=\int HdM$, $H\in L^2_{loc}(M)$, $L$ a continuous local martingale strongly orthogonal to $M$.
With strongly orthogonal we mean, that $LM$ is a local martingale. $L^2_{loc}(M)$ is the space of all predictable processes, which are locally in $L^2(M)$, i.e. there exists a sequence of stopping times such that $E[\int_0^{\tau_n}H^2 d\langle M\rangle]<\infty$.
Can we drop the continuity assumption on $N$? Of course we would then also have a general local martingale $L$. If so, can someone give me a reference, please?
Yes, there is this very nice paper. Have a look at case 3, I sum this up:
$N$ is a general local martingale. Write $N=N_0+N^c+N^d$, where $N^c$ is continuous and $N^d$ is purely discontinuous, i.e. $N^dL$ is a local martingale for every continuous local martingale $L$. By definition this means, that $N^d$ is orthogonal to every continuous local martingale. Such a decomposition exists, see e.g. "Limit Theorems For Stochastic Processes" by Jacod/Shirajev. To the continuous part $N^c$ we can apply the usual Kunita-Watanabe, since it is square integrable (see the first case in the paper). So we get
$$N^c=H\bullet M+U$$
where $M$ is your continuous local martingale, $H\in L_{loc}(M)$ and a local martingale $U$ orthogonal to $M$. Now define $L:=U+N^d$. Hence
$$N=H\bullet M+L$$
where $L$ is orthogonal to $M$, since $U$ is by the continuous case and $N^d$ since it is orthogonal to every continuous local martingale.