Definition 3.1.8 (Strong Solution).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 a mapping

is a strong solution to the SDE

if

1. (1)

   For each $t \ge 0$, let

   \[\mathcal{F}_{t} = \sigma(\bracs{\pi_s: C([0, \infty); \real^d) \to \real^d|0 \le s \le t}) \quad \mathcal{G}_{t} = \sigma(\bracs{\pi_s: C([0, \infty); \real^n) \to \real^n|0 \le s \le t})\]

   
   

   Let $\wien^{d}$ and $\wien^{n}$ be the classical Wiener measure on $C([0, \infty); \real^{d})$ and $C([0, \infty); \real^{n})$, respectively, and $\bracsn{\ol{\mathcal{G}}_t|t \ge 0}$ be the $\wien^{n}$-augmentation of $\bracs{\mathcal{G}_t|t \ge 0}$, then

   \[F^{-1}(\mathcal{F}_{t}) \subset \mathcal{B}(\real^{n}) \times \overline{\mathcal{G}_t}\]

   
   

   for all $t \ge 0$.

2. (2)

   For any filtered probability space $(\Omega, \bracs{\mathcal{H}_t}, \bp)$, random variable $\xi: \Omega \to \real^{n}$, and $\bracs{\mathcal{H}_t}$-Brownian motion $B$, the process $X = F(\xi, B)$ solves Equation 3.3.