I am trying to understand the prove of the existence of the stochastic integral for
- a local martinglale null at $0$ and continuous, $M\in \mathcal{M}^c_{0,\text{loc}}$,
- a predictable process $H\in L^2(M):=L^2(\overline{\Omega},\mathcal{P},P_M)=\{H:\parallel H\parallel_{L^2(M)}=\mathbb{E}\left[\int_0^\infty H^2_s d[M]_s\right]<\infty\}$ where $\overline{\Omega}=\Omega\times (0,\infty)$ and $\mathcal{P}$ is the $\sigma$-field on $\overline{\Omega}$ generated by all adapted left-continuous processes. Predictable processes are measurable wrto this $\sigma$-field.
- $\mathcal{H}^{2,c}_0=\{M\ \text{continuous martingale}:M_0=0\ \text{and}\ \sup_{t\geq0}\mathbb{E}[M_t^2]<\infty\}$
Theorem Fix $M\in \mathcal{M}^c_{0,\text{loc}}$. Then for each $H\in L^2(M)$, there exists a unique element $H\cdot M \in\mathcal{H}^{2,c}_0$ satisfying $$[H\cdot M,N]=\int Hd[M,N]$$ for every $N\in \mathcal{M}^c_{0,\text{loc}}$. Then $H\cdot M$ is called the stochastic integral of $H$ wrto $M$. Moreover the map $L^2(M)\rightarrow \mathcal{H}^{2,c}_0, H\mapsto H\cdot M$ is a linear isometry.
The uniqueness should be fine but my problem is the existence. Using Kunitz-Watanabe we know that the linear functional $J:\mathcal{H}^{2,c}_0\rightarrow \mathbb{R},N\mapsto J(N):=\mathbb{E}\left[\int_0^\infty H_s^2d[M,N]_s\right]$ is continuous on a Hilbert space. Therefore by Riesz representation theorem we can write it using the dot product on $\mathcal{H}^{2,c}_0$, i.e. there exists an $L\in \mathcal{H}^{2,c}_0$ such that $$J(N):=\mathbb{E}\left[\int_0^\infty H_s^2d[M,N]_s\right]=(L,N)_{\mathcal{H}^{2,c}_0}=\mathbb{E}[[L,N]_\infty] \ (\star)$$ for all $N\in \mathcal{H}^{2,c}_0$
Next the poof goes on stating that we now want to prove that for each $N\in\mathcal{H}^{2,c}_0$:$[L,N]=\int H d[M,N]$ and this will be done by showing that the difference $LN-\int H d[M,N]$ is a martingale and by uniqueness of $[L,N]$ this is enough since $\int H d[M,N]$ is continuous and predictable, hence adapted.
So first of all I am not sure about the last equality at $(\star)$, in the proof there is $\mathbb{E}[L_\infty N_\infty]$ but it does not make sense to me.
And my biggest problem is the last part of the proof. Why the existence should follows by showing that this difference is a martingale?
Thanks for any help
Are you sure about your definition of $\mathcal H_0^{2,c}$? Shouldn't this be the class of $L^2$-bounded continuous martingales? This is a indeed a Hilbert space with the scalar product $\langle L,N\rangle =E[L_\infty N_\infty]$. This makes sense because $L$ and $N$ are uniformly integrable and therefore $L_\infty=\lim L_t$ exists by the martingale convergence theorem.
Concerning your second question: Existence of $H\cdot M=L$ comes from Riesz' theorem, now you have to show that $L$ has the desired property.