Lemma 3.2.2.label Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, $B$ be a $\bracs{\mathcal{F}_t}$-adapted standard Brownian motion, $\sigma$ be a $\bracs{\mathcal{F}_t}$-previsible $L(\real^{d}; \real^{n})$-valued process, $b$ be a $\bracs{\mathcal{F}_t}$-previsible $\real^{n}$-valued process, $\xi \in L^{p}(\Omega, \cf_{0}; \real^{n})$, and
\[X_{t} = \xi + \int_{0}^{t} \sigma_{s} dB_{s} + \int_{0}^{t} b_{s} ds\]
then for any $T > 0$ and $p \ge 2$, there exists $C_{T, p, n}\ge 0$ such that for all $0 \le t \le T$,
\[\ev\braks{\norm{X}_{u, [0, t]}^p}\le C_{T, p, n}\braks{\ev(\norm{\xi}_{\real^n}^p) + \ev\paren{\int_0^t \norm{\sigma_s}_{\real^n}^p + \norm{b_s}_{\real^n}^p ds }}\]
Proof, [Theorem 11.5, RW89]. Assume without loss of generality that $\xi = 0$, then
\[\norm{X}_{u, [0, t]}^{p} \le C_{p}\paren{\sup_{0 \le s \le t}\int_0^s \sigma_rdB_r}^{p} + \paren{\int_0^t \norm{b_s}_{\real^n}ds}^{p}\]
where by Jensen’s inequality,
\[\paren{\int_0^t \norm{b_s}ds}^{p} \le C_{t, p}\int_{0}^{t} \norm{b_s}_{\real^n}^{p} ds\]
and by the BDG inequality and Jensen’s inequality,
\[\ev\braks{\paren{\sup_{0 \le s \le t}\int_0^s \sigma_rdB_r}^p}\le C_{p} \ev\braks{\paren{\int_0^t \norm{\sigma_s}_{\real^d}^2 ds}^{p/2}}\le C_{p} \int_{0}^{t} \norm{\sigma_s}_{\real^d}^{p} ds\]
$\square$