3.3 Weak Solutions

Definition 3.3.1 (Weak Solution).label Let $\mu$ be a Borel probability measure on $\real^{n}$, $\sigma: [0, \infty) \times C([0, \infty); \real^{d}) \to L(\real^{d}; \real^{n})$ and $b: [0, \infty) \times C([0, \infty); \real^{d}) \to \real^{n}$ be previsible path functionals, then the SDE

has a weak solution with initial distribution $\mu$ if there exists a filtered probability space $(\Omega, \bracs{\cf_t}, \bp)$, a $\bracs{\mathcal{F}_t}$-Brownian motion $B$, and a $\bracs{\mathcal{F}_t}$-semimartingale $X: \Omega \to C([0, \infty); \real^{d})$ such that:

1. (1)

   $X_{0}$ has distribution $\mu$.

2. (2)

   For each $t > 0$,

   \[\int_{0}^{t} \norm{\sigma(s, X)}_{\real^n}^{2} + \norm{\sigma(s, X)}_{\real^n}^{2} ds < \infty\]

   
   

   almost surely.

3. (3)

   For each $t \ge 0$,

   \[X_{t} = X_{0} + \int_{0}^{t} \sigma(s, X) dB_{s} + \int_{0}^{t} b(s, X) ds\]

   
   

Definition 3.3.2 (Uniqueness in Distribution).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

has uniqueness in law if for any solutions $X$ and $X'$ such that the distributions of $X_{0}$ and $X_{0}'$ are the same, the distributions of $X$ and $X'$ are the same.

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

then the above SDE is exact if and only if the following conditions hold:

1. (1)

   The SDE has a weak solution.

2. (2)

   The SDE has the pathwise-uniqueness property.

If the above holds, then the SDE also has uniqueness in law.