Is there a Burkholder-Davis-Gundy inequality for martingale increments?

Question

is there a Burkholder-Davis-Gundy inequality for martingale increments? More specifically, I would like to find a finite bound of order $h^{p/2}$ for the expectation $$\operatorname{E} \left[ \sup_{t \geq 0} \left| \int_t^{t+h} X_s \, \mathrm{d} B_s \right|^p \right]$$ where $B$ is a Brownian motion and $\operatorname{E} \left[ \sup_{s \geq 0} |X_s|^p \right]< \infty$. If the increment would be fixed, one could apply Burkholder-Davis-Gundy, but unfortunately this is not the case. Is there some kind of standard method to handle this case or even a Burkholder-Davis-Gundy inequality for increments?