Recently I'm reading Jacod's Limit Theorems for Stochastic Process ,chapter 1 and I'm confused with the general stochastic integral for semimartingales.

$H$ is locally bounded predictable process.

semimartingales $X$=local martingale $M$+finite variation process $A$.

$H\cdot A$ is easy to define.

$H\cdot M$ is easy to define (by Dolean's measure) if $M$ is a square integrable martingale ($\sup_t\mathbb EM_t^2<\infty$),then by localization we can define for locally square integrable $M$.

but local martingale $M$=locally square integrable $M'$+finite variation process $A'$.(proved by compensators which is hard)

so we can define stochastic integral for semimartingales.

Q1:what's the usage of the decomposition of $\mathcal{M}_0^2$(square integrable martingale space ) since every book concerning general theory mentioned this?


I know it is much more easier to define stochastic integral for $\mathcal{M}_0^{2c}$,because the Doob-Meyer decomposition is easier in this case. Moreover , in Rogers Diffusions Markov Process and Martingales , he said

the crucial point is finite variation martingales are dense in $\mathcal{M}_0^{2d}$

Q2:why it is crucial? I think the crucial point is the decomposition of local martingales.

Is there any relation between the property above and the decomposition of local martingales?

Thanks a lot !


Q1: A considerable part of the later development of stochastic integration (see chapter 2) is based on the fact that for a given optional process whose predictable projection is zero (and who satisfies an additional integrability requirement) one can find a purely discontinuous local martingale whose jump process is indistinguishable from the given optional process.


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