The Girsanov theorem

Whether a filtration satisfies the usual conditions depends on the class of null sets and hence the probability measure involved matters. Suppose $\left\{ \mathcal{F}_{t}\right\}$ satisfies the usual conditions with respect to $\mathbb {P}, H$ is a bounded predictable process, $W$ a Brownian motion with respect to $\mathbb {P}, M$ defined by (13.3), and $\mathbb {Q}$ defined by (13.4). If $t_{0}>0$ and $A \in \sigma \left(W_{s} ; s \leq t_{0}\right)$, show $\mathbb {P}\left(A\right)=0$ if and only if $\mathbb {Q}(A)=0$.

Theorem 9.10 allows us to avoid some calculations in the last paragraph of the proof of Theorem 13.3. Suppose $X$ is a continuous semimartingale under $\mathbb {P}$ and $\mathbb {Q}$ is a probability measure equivalent to $\mathbb {P}$. That is, a set is a null set for $\mathbb {P}$ if and only if it is a null set for $\mathbb {Q}$. Show $X$ is a semimartingale under $\mathbb {Q}$ and the quadratic variation of $X$ under $\mathbb {P}$ equals the quadratic variation of $X$ under $\mathbb {Q}$.

Let $W=\left(W^{1}, \ldots , W^{d}\right)$ be a $d$-dimensional Brownian motion with minimal augmented filtration $\left\{ \mathcal{F}_{t}\right\}$ and let $H_{1}, \ldots , H_{d}$ be bounded predictable processes. Let

Define a probability measure $\mathbb {Q}$ by setting $\mathbb {Q}(A)=\mean_{\mathbb {P}}\left[M_{t} ; A\right]$ if $A \in \mathcal{F}_{t}$. Let $\widetilde{W}_{t}^{i}=W_{t}^{i}-$ $\int_{0}^{t} H_{i}(s) d s$ for each $i$. Prove that $\widetilde{W}=\left(\widetilde{W}^{1}, \ldots , \widetilde{W}^{d}\right)$ is a $d$-dimensional Brownian motion under $\mathbb {Q}$.

Let $W_{t}$ be a $d$-dimensional Brownian motion and let $\delta , t_{0}>0$. Let $f:\left[0, t_{0}\right] \rightarrow \mathbb {R}^{d}$ be a continuous function. Prove that there exists a constant $c$ such that

This is known as the support theorem for Brownian motion.

Hint: First assume that $f$ has a bounded derivative. Use Exercise 4.9 and the Girsanov theorem.

Here is a more general form of the Girsanov theorem. Suppose $L_{t}$ is a bounded continuous martingale under $\mathbb {P}$, $M_{t}=e^{L_{t}-\langle L\rangle_{t}/2}$, and $\mathbb {Q}$ is a probability measure defined by $\mathbb {Q}(A)=\mathbb {E}_{\mathbb {P}}\left[M_{t_{0}} ; A\right]$ if $A \in \mathcal{F}_{t_{0}}$. Suppose $\left\{ \mathcal{F}_{t}\right\}$ is a filtration satisfying the usual conditions with respect to both $\mathbb {P}$ and $\mathbb {Q}$. Show that if $X$ is a martingale under $\mathbb {P}$, then $X_{t}-\langle X, L\rangle_{t}$ is a martingale under $\mathbb {Q}$.