Theorem 3.2.4 (Itô).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. If for each $T \ge 0$,
then the SDE
is exact and pathwise unique.
Proof of existence, [Theorem 11.2, RW89]. For each $t \ge 0$, let $\mathcal{G}_{t}^{\circ} = \sigma(\bracs{B_s|1 \le s \le t}\cup \bracs{\xi})$, $\mathcal{G}^{\circ} = \sigma(\bracsn{\mathcal{G}_t^\circ|t \ge 0})$, $\mathcal{N}$ be the collection of $\bp$-null sets in the completion of $\sigma(\bracsn{\mathcal{G}_t^\circ|t \ge 0})$, and $\mathcal{G}_{t} = \sigma(\mathcal{G}_{t}^{\circ} \cup \mathcal{N})$.
Let $\xi \in L^{\infty}(\Omega, \cf_{0}; \real^{n})$ and $X^{(0)}= \xi$. For each $m \in \natz$, define inductively
then $X^{(m)}$ has continuous sample paths, and for each $T > 0$, $\ev[{\normn{X^{(m)}}_{u, [0, s]}^2}] < \infty$. By Lemma 3.2.3,
Therefore
and $X^{(m)}$ converges uniformly in $L^{2}$ to a limiting process $X$, which satisfies the given equation.$\square$
Proof of uniqueness. Let $X$ and $Y$ be solutions of the SDE with the same setup, then by Lemma 3.2.3,
which implies that $X|_{[0, T]}= Y|_{[0, T]}$ almost surely.$\square$