Theorem 5.1.2 ([Theorem 38.9, RW89]).label Let $\wien$ be the classical Wiener measure on $C([0, \infty); \real^{d})$ and $\bracs{X_t|t \ge 0}$ be the canonical process. For each $t \ge 0$, let $\mathcal{F}_{t} = \sigma(\bracs{X_s|0 \le s \le t})$, then for any $\bracs{\mathcal{F}_t}$-previsible process $\bracs{c_t|t \ge 0}$ such that
\[\zeta_{t} = \exp\braks{\int_0^t c_sdX_s - \frac{1}{2}\int_0^t |c_s|^2ds}\]
is a martingale, then
- (1)
There exists a unique measure $\mathcal{V}$ on $(\Omega, \mathcal{F}_{\infty})$ such that for each $t \ge 0$,
\[\frac{d \mathcal{V}}{d \mathcal{W}}\bigg | \cf_{t^+}= \zeta_{t}\] - (2)
The process
\[\tilde X_{t} = X_{t} - \int_{0}^{t} \gamma_{s} ds\]is a $\bracs{\mathcal{F}_{t^+}}$-Brownian motion.