Lemma 3.2.3.label Let $\sigma: [0, \infty) \times C([0, \infty); \real^{d}) \to L(\real^{d}; \real^{n})$ and $b: [0, \infty) \times C([0, \infty); \real^{d}) \to \real^{n}$ be previsible path functionals satisfying the Lipschitz condition with constant $K$.
Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space and $B$ be a $\bracs{\mathcal{F}_t}$-adapted standard Brownian motion. For any $\bracs{\mathcal{F}_t}$-adapted process $X: \Omega \to C([0, \infty); \real^{d})$ with continuous sample paths and $\xi \in L^{p}(\Omega, \cf_{0}; \real^{n})$, let
\[P(X, \xi)_{t} = \xi + \int_{0}^{t} \sigma(s, X)dB_{s} + \int_{0}^{t} b(s, X)ds\]
then for any $\bracs{\mathcal{F}_t}$-adapted process $Y: \Omega \to C([0, \infty); \real^{d})$, $\eta \in L^{p}(\Omega, \cf_{0}; \real^{n})$, $T > 0$, and $0 \le t \le T$,
\begin{align*}&\ev\braks{\norm{P(X, \xi) - P(Y, \eta)}_{u, [0, t]}^p}\\&\le C_{K, n, T, p}\braks{\norm{\xi - \eta}_{L^p(\Omega; \real^n)}^p + \ev\paren{\int_0^t \norm{X - Y}_{u, [0, s]}^p ds}}\end{align*}