Let $T>0$, $F$ be an adapted squared-integrable process and $p\ge 2$, then the Burkholder's inequality implies $$ E\bigg(\bigg|\int_0^T F(s)dB(s)\bigg|^p\bigg)\le C(p) E\bigg[\bigg(\int_0^T F^2(s)ds\bigg)^{p/2}\bigg], $$ for some constant $C(p)$.
I was wondering why most textbooks present the inequality from $[0,T]$, instead of any subinterval $[s,t]\subset[0,T]$. I follow the proof and I didn't notice any technical difficulty to extend the result to the integral on $[s,t]$: $$ E\bigg(\bigg|\int_s^t F(r)dB(r)\bigg|^p\bigg)\le C(p) E\bigg[\bigg(\int_s^t F^2(r)dr\bigg)^{p/2}\bigg]. $$ Did I overlook anything? Or the result on $[s,t]$ is a simple consequence of the result on $[0,t]$?
Similarly, I was wondering whether the $L^p$ estimate of Levy integral: $$E\bigg(\bigg|\int_0^T\int_E F(s)d\tilde{N}(ds,de)\bigg|^p\bigg),$$ the so-called Kunita's inequality, works on arbitrary fixed finite interval $[s,t]$.