5.1 Girsanov Transforms

Theorem 5.1.1 (Cameron-Martin-Girsanov, [Theorem 38.5, RW89]).label Let $\wien$ be the classical Wiener measure on $C([0, \infty); \real^{d})$ and $\mathcal{V}$ be equivalent to $\wien$. For each $t \ge 0$, let $\bracs{X_t \ge 0}$ be the canonical process, $\mathcal{F}_{t}^{\circ} = \sigma(\bracs{X_s|0 \le s \le t})$, $\bracs{\mathcal{F}_t}$ be its $\wien/\mathcal{V}$-augmentation, then there exists a previsible $\real^{n}$-valued process $\bracs{c_t|t \ge 0}$ such that for each $t \ge 0$,

and

is a Brownian motion under $\mathcal{V}$.

Conversely, for any previsible $\real^{n}$-valued process $\bracs{c_t|t \ge 0}$,

is a uniformly integrable martingale with respect to

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

is a martingale, then

1. (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. (2)

   The process

   \[\tilde X_{t} = X_{t} - \int_{0}^{t} \gamma_{s} ds\]

   
   

   is a $\bracs{\mathcal{F}_{t^+}}$-Brownian motion.