Definition 3.1.7 (Pathwise Exact).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, then the SDE
\[X_{t} = \xi + \int_{0}^{t} \sigma(s, X) dB_{s} + \int_{0}^{t} b(s, X) ds\]
is pathwise exact if given
A filtered probability space $(\Omega, \bracs{\cf_t}, \bp)$,
$\bracs{\mathcal{F}_t}$-adapted standard Brownian motion $B$,
$\bracs{\mathcal{F}_t}$-semimartingales $X, Y: \Omega \to C([0, \infty); \real^{d})$ satisfying Equation 3.2 and Equation 3.1,
then for every $t \ge 0$, $X_{t} = Y_{t}$ almost surely.