Theorem 3.3.3 (Yamada and Watanabe, [Theorem 17.1, RW89]).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, and
\[X_{t} = X_{0} + \int_{0}^{t} \sigma(s, X) dB_{s} + \int_{0}^{t} b(s, X) ds\]
then the above SDE is exact if and only if the following conditions hold:
- (1)
The SDE has a weak solution.
- (2)
The SDE has the pathwise-uniqueness property.
If the above holds, then the SDE also has uniqueness in law.