Theorem 3.4.2 (Equivalence of Formulations).label Let $(\Omega, \bracsn{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, $B$ be a $\bracs{\mathcal{F}_t}$-Brownian motion, $\sigma: [0, \infty) \times C([0, \infty); \real^{n}) \to L(\real^{n}; \real^{n})$ and $b: [0, \infty) \times C([0, \infty); \real^{n}) \to \real^{n}$ be bounded measurable functions such that $\sigma_{t}$ is invertible for all $t \ge 0$.

Let $y \in \real^{n}$ and $X: \Omega \to C([0, \infty); \real^{n})$ be a solution to the martingale problem for $(\sigma^{*}\sigma, b)$ starting at $y$, then there exists a weak solution of the SDE

starting at $y$ whose distribution is the same as $X$.