I want to prove these inequalities, i.e.:
For $p\geq 1\ \exists 0<c_p\leq C_p$ such that for any martingale $M$ we have the following inequality: $$c_p\mathbb{E}[[M,M]^{p/2}_\infty]\leq \mathbb{E}[(\mid M\mid^*_\infty)^p]\leq C_p\mathbb{E}[[M,M]^{p/2}_\infty]$$ holds true.
The proof I have want to derive it from the same inequality for discrete martingales. It states that inequalities of the following type hold: $$(h\bullet M)_T+[M,M]^{p/2}_T\leq(\mid M\mid^*_T)^p\leq C_p[M,M]^{p/2}_T+(g\bullet M)_T$$ for predictable integrands $h,g$, martingale M and upper bound $T$ (note that $\mid M\mid^*_t:=\sup_{0\leq s\leq t}{M_s}$).
Now assuming the above, we have the following fact:
- We can choose a sequence of partitions $\sigma_n$ tending to the identity (i.e. the interval $[0,\infty)$) such that $M^{\sigma _n}\rightarrow M$ in (UCP)
- From the above fact follows that $[M^{\sigma_n},M^{\sigma_n}]\rightarrow [M,M]$ in (UCP)
The sequel isn't that clear to me. We restrict on the case where $\mathbb{E}[(\mid M\mid^*_\infty)^p]<\infty$ otherwise we get infinity. Then he states that we have convergence $M^{\sigma_n}\rightarrow M$ but not what kind of convergence. and moreover that $$\mathbb{E}[\mid[M^{\sigma_m},M^{\sigma_m}]^{1/2}_T-[M^{\sigma_n},M^{\sigma_n}]^{1/2}_T\mid^p]\mathbb{E}[[M^{\sigma_m}-M^{\sigma_n},M^{\sigma_m}-M^{\sigma_n}]^{p/2}_T]\leq\mathbb{E}[(\mid M^{\sigma_m}-M^{\sigma_n}\mid^*_T)^p]\rightarrow 0$$ And so we have $L^p$ convergence of the quadratic variations to $[M,M]$. And as it holds for any $T>0$ we can let $T\rightarrow \infty$.
For the last steps I would need some help because I don't see what he want to do and no more details are given.
Thanks