Let $(\Omega,\mathcal{F},\mathbb{F}=(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be a stochastic basis and $M$ a continuous local martingale on it with start in $0$. Then the Burkholder-Davis-Gundy inequality states that for all $p>0$ there exists some $C_p>0$ s.t. $$ \mathbb{E}[\sup_{s \leq t}|M_s|^p] \leq C_p\mathbb{E}[\langle M\rangle_t ^{\frac{p}{2}} ] $$
holds for all $t \geq 0$.
If $M$ is multidimensional(i.e. every component is a continuous local martingale starting in $0$), does
$$ \mathbb{E}[\sup_{s \leq t}\|M_s\|_p^p] \leq C_p\mathbb{E}[\|\langle M\rangle_t \|_p^{\frac{p}{2}} ] $$
hold? It would be very useful to me but I can not find this anywhere and I was not able to proof it. I would be grateful for a reference or any hint.