Etienne Adolphe Piot (1825-1910) was a French painter.
The topics are as follows
\begin{equation}{\label{a}}\tag{A}\mbox{}\end{equation}
Change of Time.
We shall prove that any continuous local martingale \(M\) can be time-changed to a Brownian motion run up to the time \(\langle M\rangle_{\infty}\). Let \(\{M_{t}\}_{t\geq 0}\) be a continuous local martingale
with respect to the filtration \(\{{\cal F}_{t}\}_{t\geq 0}\). Define
\[\langle M\rangle_{\infty}=\sup_{t\geq 0}\langle M\rangle_{t}.\]
Then since \(\langle M\rangle\) is \(P\)-a.s. increasing, \(\langle M\rangle_{\infty}=\lim_{t\rightarrow\infty}\langle M\rangle_{t}\) \(\mathbb{P}\)-a.s. For each \(t\geq 0\), let
\[T_{t}=\inf\{s\geq 0:\langle M\rangle_{s}>t\}.\]
Recall that by convention \(\inf\emptyset =+\infty\). We first consider the case where \(\langle M\rangle_{\infty}=\infty\) \(P\)-a.s. In this case, \(M\) can be time-changed to a Brownian motion run for all time.
Theorem. (Chung and Williams \cite{chu}). For the continuous local martingale \(\{M_{t}\}_{t\geq 0}\) with respect to the filtration \(\{{\cal F}_{t}\}_{t\geq 0}\), suppose that $latex \langle
M\rangle_{\infty}=\infty$ \(P\)-a.s. Then \(\{M_{T_{t}}\}_{t\geq 0}\) is indistinguishable from a Brownian motion in \(\mathbb{R}\). \(\sharp\)
We now consider the general case where \(\langle M\rangle_{\infty}\) may be finite with positive probability. For this we introduce a Brownian motion independent of \(M\) which will be used to continue the time-changed version of \(M\) to a Brownian motion run for all time. Let \(\widehat{W}\) be a one-dimensional Brownian motion defined on a probability space \((\widehat{\Omega},\widehat{{\cal F}},\widehat{\mathbb{P}})\) that is independent of \((\Omega ,{\cal F},P)\) and suppose \(\widehat{W}_{0}=0\). Define \((\widehat{\Omega},\widehat{{\cal F}},\widehat{\mathbb{P}})\) to be the completion of \((\Omega\times\widehat{\Omega},{\cal F}\times\widehat{{\cal F}},P\times\widehat{\mathbb{P}})\). In the following, two sets are almost surely equal if their symmetric difference has probability zero.
Theorem. For the continuous local martingale \(\{M_{t}\}_{t\geq 0}\) with respect to \(\{{\cal F}_{t}\}_{t\geq 0}\), we have \(P\)-a.s.
\begin{equation}{\label{chueq916}}
\left\{\lim_{t\rightarrow\infty}M_{t}\mbox{ exists and is finite}\right\}=\left\{\langle M\rangle_{\infty}<\infty\right\}
\end{equation}
and
\[\limsup_{t\rightarrow\infty}\frac{M_{t}}{\sqrt{2\cdot\langle M\rangle_{t}\cdot\log\log\langle M\rangle_{t}}}=1\mbox{ on }\left\{\langle M\rangle_{\infty}=\infty\right\}.\]
Let
\[\Gamma =\left\{\langle M\rangle_{\infty}<\infty\mbox{ and }\lim_{t\rightarrow\infty}M_{t}\mbox{ does not exist in }\mathbb{R}\right\},\]
which is a \(\mathbb{P}\)-null set by \((\ref{chueq916})\). Let \(T_{t}=\inf\{ s\geq 0:\langle M\rangle_{s}>t\}\) and define \(M_{\infty}=\lim_{t\rightarrow\infty}M_{t}\), wherever this limit exists and is finite. For each \((t,\omega , \widehat{\omega})\in\mathbb{R}_{+}\times\Omega\times\widehat{\Omega}\), let
\begin{equation}{\label{chueq918}}
X_{t}(\omega ,\widehat{\omega})=\left\{\begin{array}{ll}
M_{T_{t}}(\omega )+\left (\widehat{W}_{t}(\widehat{\omega})-
\widehat{W}_{\langle M\rangle_{\infty}(\omega )}(\widehat{\omega})\right )
\cdot 1_{\{\langle M\rangle_{\infty}(\omega )<t\}} &
\mbox{if \(\omega\in\Omega\setminus\Gamma\)}\\
0 & \mbox{if \(\omega\in\Gamma\)}.
\end{array}\right .
\end{equation}
Then \(X\) on \((\widehat{\Omega},\widehat{{\cal F}},\widehat{\mathbb{P}})\) is indistinguishable from a Brownian motion in \(\mathbb{R}\). Note that for \(\widehat{\mathbb{P}}\)-a.e. \((\omega ,\widehat{\omega})\in\widehat{\Omega}\), \(X_{t}(\omega ,\widehat{\omega})\) is defined by the first expression in \((\ref{chueq918})\). \(\sharp\)
\begin{equation}{\label{b}}\tag{B}\mbox{}\end{equation}
Change of Measure and Girsanov’s Theorem.
In the sequel, we consider the behavior of local martingales under mutually absolutely continuous changes of probability measure. When applied to solutions of stochastic differential equations, these results can be used to change the drift, simply by changing the ambient probability measure in a mutually absolutely continuous manner over each “finite time horizon”, i.e. on the \(\sigma\)-fields generated by the process on each finite time interval.
The following discussion follows from Klebaner \cite{kle}. Suppose that a random variable \(X\) has standard normal \(N(0,1)\) distribution. This presumes a probability measure \(P\) under which \(X\) is \(N(0,1)\). Take any \(\mu\neq 0\). Let \(f_{0}(x)\) denote the density of \(N(0,1)\) distribution, and \(f_{\mu}(x)\) denote the density of \(N(\mu ,1)\) distribution
\[f_{\mu}(x)=\frac{1}{\sqrt{2\pi}}\cdot e^{-\frac{1}{2}(x-\mu )^{2}}=f_{0}(x)\cdot e^{\mu x-\mu^{2}/2}.\]
By the definition of a density function, the probability of a set \(A\) on the line, is the integral of the density over this set,
\[P_{X}(A)=P\{X\in A\}=\int_{A}f_{0}(x)dx=\int_{A}d\mathbb{P}_{X}.\]
In infinitesimal notations this relation is often written as
\[d\mathbb{P}_{X}=P_{X}(dx)=P\{X\in dx\}=f_{0}(x)dx.\]
Define the new probability measure \(P_{\mu}\) by
\[d\mathbb{P}_{\mu}=e^{\mu X-\mu^{2}/2}d\mathbb{P}.\]
Then we have
\[P_{\mu}\{X\in A\}=\int_{A}d\mathbb{P}_{\mu}=\int_{A}e^{\mu X-\mu^{2}/2}d\mathbb{P}=\int_{A}e^{\mu x-\mu^{2}/2}f_{0}(x)dx=\int_{A}f_{\mu}(x)dx.\]
Thus, under \(P_{\mu}\), \(X\) is \(N(\mu ,1)\). The quantity
\[\Gamma (x)=\frac{P_{\mu}\{X\in dx\}}{P\{X\in dx\}}=\frac{f_{\mu}(x)}{f_{0}(x)}=e^{\mu x-\mu^{2}/2}\]
is known as the Likelihood ratio. Therefore we can write \(d\mathbb{P}_{\mu}=\Gamma (X)d\mathbb{P}\).
Let us consider independent normal random variables \(Z_{1},\cdots ,Z_{n}\) on \((\Omega ,{\cal F},P)\) with \(E[Z_{i}]=0\) and \(\mathbb{E}[Z_{i}^{2}]=1\) (i.e \(Z_{i}\) are \(N(0,1)\) for all \(i\)). Given a
vector \((\mu_{1},\cdots ,\mu_{n})\in \mathbb{R}^{n}\), we consider the new probability measure \(\widehat{\mathbb{P}}\) on \((\Omega ,{\cal F})\) given by
\[\widehat{\mathbb{P}}(d\omega )=\exp\left (\sum_{i=1}^{n}\mu_{i}\cdot Z_{i}(\omega )-\frac{1}{2}\sum_{i=1}^{n}\mu_{i}^{2}\right )\cdot P(d\omega ).\]
Then
\begin{align*}
\widehat{\mathbb{P}}\left\{Z_{1}\in dz_{1},\cdots ,Z_{n}\in dz_{n}\right\} & =\exp\left (\sum_{i=1}^{n}\mu_{i}\cdot z_{i}-\frac{1}{2}\sum_{i=1}^{n}
\mu_{i}^{2}\right )\cdot P\left\{Z_{1}\in dz_{1},\cdots ,Z_{n}\in dz_{n}\right\}\\
& =\exp\left (\sum_{i=1}^{n}\mu_{i}\cdot z_{i}-\frac{1}{2}\sum_{i=1}^{n}\mu_{i}^{2}\right )\cdot f(z_{1},\cdots ,z_{n})dz_{1}\cdots dz_{n}\\
& =\exp\left (\sum_{i=1}^{n}\mu_{i}\cdot z_{i}-\frac{1}{2}\sum_{i=1}^{n}\mu_{i}^{2}\right )\cdot (2\pi )^{-n/2}\cdot\exp\left (-\frac{1}{2}\sum_{i=1}^{n}z_{i}^{2}\right )dz_{1}\cdots dz_{n}\\
& =(2\pi )^{-n/2}\cdot\exp\left (-\frac{1}{2}\sum_{i=1}^{n}(z_{i}-\mu_{i})^{2}\right )dz_{1}\cdots dz_{n}.
\end{align*}
Therefore, under \(\widehat{\mathbb{P}}\) the random variables \(Z_{1},\cdots ,Z_{n}\) are independent and normal with \(\mathbb{E}_{\widehat{\mathbb{P}}}[Z_{i}]=\mu_{i}\) and \(\mathbb{E}_{\widehat{\mathbb{P}}}[(Z_{i}-\mu_{i})^{2}]=1\). In other words, \(\{\widehat{Z}_{i}=Z_{i}-\mu_{i}\}_{i=1}^{n}\) are independnet standard normal random variables \(N(0,1)\) on \((\Omega ,{\cal F},\widehat{\mathbb{P}})\). The Girsanov theorem extends this idea of invariance of Gaussian finite-dimensional distributions under appropriate translations and changes of the underlying probability measure, from the discrete to the continuous setting.
Definition. Two probability measures \(\mathbb{P}\) and \(\mathbb{Q}\) on \((\Omega ,{\cal F})\) are said to be equivalent if \(\mathbb{P}\ll \mathbb{Q}\) and \(\mathbb{Q}\ll\mathbb{P}\). \(\sharp\)
Theorem. (Radon-Nikodym). Let \(\mathbb{Q}\ll \mathbb{P}\), then there exists a random variable \(\Gamma\) such that \(\Gamma\geq 0\), \(\mathbb{E}_{\mathbb{P}}[\Gamma ]=1\), and
\begin{equation}{\label{kleeq108}}
Q(A)=\mathbb{E}_{\mathbb{P}}[\Gamma\cdot 1_{A}]=\int_{A}\Gamma d\mathbb{P}
\end{equation}
for any measurable set \(A\). \(\Gamma\) is \(\mathbb{P}\)-a.s. unique. Conversely, if there exists a \(\Gamma\) with the above properties and \(\mathbb{Q}\) is defined by (\ref{kleeq108}), then it is a probability measure and \(\mathbb{Q}\ll \mathbb{P}\). \(\sharp\)
The random variable \(\Gamma\) is also denoted by \(d\mathbb{Q}/d\mathbb{P}\). It follows from (\ref{kleeq108}) that if \(Q\ll P\), then expectations under \(\mathbb{P}\) and \(\mathbb{Q}\) are related by
\begin{equation}{\label{kleeq109}}
\mathbb{E}_{\mathbb{Q}}[Z]=\mathbb{E}_{\mathbb{P}}[\Gamma Z]\mbox{ or }\int Zd\mathbb{Q}=\int \Gamma Zd\mathbb{P}
\end{equation}
for any random variable \(Z\) integrable with resoect to \(\mathbb{Q}\).
\begin{equation}{\label{klet103}}\tag{1}\mbox{}\end{equation}
Proposition \ref{klet103}. (Klebaner \cite{kle}). Let \(X\) have \(N(0,1)\) distribution under \(\mathbb{P}\) and let \(Y=X+\mu\). Then, there is an equivalent probability measure \(\mathbb{Q}\) such that \(Y\) is \(N(0,1)\) under \(\mathbb{Q}\). The likelihoods are given by
\[\frac{d\mathbb{Q}}{d\mathbb{P}}=e^{-\mu X-\mu^{2}/2}\mbox{ and }\frac{d\mathbb{P}}{d\mathbb{Q}}=e^{\mu X+\mu^{2}/2}=e^{\mu Y-\mu^{2}/2}. \sharp\]
Note that under the original probability measure \(\mathbb{P}\), \(Y=X+\mu\) has mean \(\mu\), but under \(\mathbb{Q}\) it has mean zero, therefore this is way to “remove the mean”.
\begin{equation}{\label{klet105}}\mbox{}\end{equation}
Proposition \ref{klet105}. (Klebaner \cite{kle}). Let \({\cal G}\) be a sub-$\sigma$-field of \({\cal F}\) on which two probability measures \(\mathbb{P}\) and \(\mathbb{Q}\) are given. If \(\mathbb{Q}\ll \mathbb{P}\) with \(d\mathbb{Q}=\Gamma d\mathbb{P}\) and \(X\) is \(\mathbb{Q}\)-integrable, then \(\Gamma X\) is \(\mathbb{P}\)-integrable and
\begin{equation}{\label{kleeq1011}}
\mathbb{E}_{\mathbb{Q}}[X|{\cal G}]=\frac{\mathbb{E}_{\mathbb{P}}[X\Gamma |{\cal G}]}{\mathbb{E}_{\mathbb{P}}[\Gamma |{\cal G}]}\mbox{ \(\mathbb{Q}\)-a.s.}
\end{equation}
Proof. Note that
\begin{align*}
Q\left\{\mathbb{E}_{\mathbb{P}}[\Gamma |{\cal G}]=0\right\} & =\int_{\{\mathbb{E}_{\mathbb{P}}[\Gamma |{\cal G}]=0\}}d\mathbb{Q}=\int_{\{\mathbb{E}_{\mathbb{P}}[\Gamma |{\cal G}]=0\}}
\Gamma d\mathbb{P}=\mathbb{E}_{\mathbb{P}}\left [\Gamma\cdot 1_{\{\mathbb{E}_{\mathbb{P}}[\Gamma |{\cal G}]=0\}}\right ]\\
& =\mathbb{E}_{\mathbb{P}}\left .\left [\mathbb{E}_{\mathbb{P}}\left [\Gamma\cdot 1_{\{\mathbb{E}_{\mathbb{P}}[\Gamma |{\cal G}]=0\}}\right ]\right |{\cal G}\right ]=\mathbb{E}_{\mathbb{P}}\left [\mathbb{E}_{\mathbb{P}}[\Gamma |{\cal G}]\cdot 1_{\{\mathbb{E}_{\mathbb{P}}[\Gamma |{\cal G}]=0\}}\right ]\\
& =\int_{\Omega}\mathbb{E}_{\mathbb{P}}[\Gamma |{\cal G}]\cdot 1_{\{\mathbb{E}_{\mathbb{P}}[\Gamma |{\cal G}]=0\}}d\mathbb{P}=0
\end{align*}
so that (\ref{kleeq1011}) is well-defined. The integrability of \(\Gamma X\) follows from
\[\mathbb{E}_{\mathbb{P}}[\Gamma X|]=\mathbb{E}_{\mathbb{P}}[\Gamma |X|]=\int_{\Omega}\Gamma Xd\mathbb{P}=\int_{\Omega}|X|d\mathbb{Q}=\mathbb{E}_{\mathbb{Q}}[|X|]<\infty .\]
Recall that \(Q\left\{\mathbb{E}_{\mathbb{P}}[\Gamma |{\cal G}]=0\right\}=0\), so that \(\frac{\mathbb{E}_{\mathbb{P}}[X\Gamma |{\cal G}]}{\mathbb{E}_{\mathbb{P}}[\Gamma |{\cal G}]}\) is well-defined and is \({\cal G}\)-measurable. Now for any \(A\in {\cal G}\), we have
\begin{align*}
\mathbb{E}_{\mathbb{Q}}\left [\frac{\mathbb{E}_{\mathbb{P}}[X\Gamma |{\cal G}]}{\mathbb{E}_{\mathbb{P}}[\Gamma |{\cal G}]}\cdot 1_{A}\right ] & = & \mathbb{E}_{\mathbb{P}}\left [\Gamma\cdot\frac{\mathbb{E}_{\mathbb{P}}[X\Gamma |{\cal G}]}{\mathbb{E}_{\mathbb{P}}[\Gamma |{\cal G}]}\cdot 1_{A}\right ]\mbox{ (by (\ref{kleeq109}))}\\
& = & \mathbb{E}_{\mathbb{P}}\left .\left [\mathbb{E}_{\mathbb{P}}\left [\Gamma\cdot \frac{\mathbb{E}_{\mathbb{P}}[X\Gamma |{\cal G}]}{\mathbb{E}_{\mathbb{P}}[\Gamma |{\cal G}]}\cdot 1_{A}\right ]\right |{\cal G}\right ]=\mathbb{E}_{\mathbb{P}}\left [\mathbb{E}_{\mathbb{P}}[\Gamma |{\cal G}]\cdot\frac{\mathbb{E}_{\mathbb{P}}[X\Gamma |{\cal G}]}{\mathbb{E}_{\mathbb{P}}[\Gamma |{\cal G}]}\cdot 1_{A}\right ]\\
& = & \mathbb{E}_{\mathbb{P}}\left [\mathbb{E}_{\mathbb{P}}[X\Gamma |{\cal G}]\cdot 1_{A}\right ]=\mathbb{E}_{\mathbb{P}}[X\Gamma\cdot 1_{A}]=\mathbb{E}_{\mathbb{Q}}[X\cdot 1_{A}],
\end{align*}
that is, for any \(A\in {\cal G}\),
\[\int_{A}Xd\mathbb{Q}=\int_{A}\frac{\mathbb{E}_{\mathbb{P}}[X\Gamma |{\cal G}]}{\mathbb{E}_{\mathbb{P}}[\Gamma |{\cal G}]}d\mathbb{Q}.\]
Then (\ref{kleeq1011}) follows immediately. \(\blacksquare\)
There is a change of measure for Brownian motion with drift that makes it into Brownian motion without drift, similar to the removal of the mean for Gaussian random variables in Proposition~\ref{klet103}. First
we give a general result for calculation of expectations and conditional expectations under a change of measure which follows directly from Proposition \ref{klet105}.
Proposition. (Klebaner \cite{kle}). Let \(\{\Gamma_{s}\}_{0\leq s\leq t}\) be a positive \(P\)-martingale such that \(\mathbb{E}_{\mathbb{P}}[\Gamma_{t}]=1\). Define the new probability measure \(Q\) by the relation
\[Q(A)=\int_{A}\Gamma_{t}d\mathbb{P},\]
denoted by \(d\mathbb{Q}/d\mathbb{P}=\Gamma_{t}\). Then \(Q\) is absolutely continuous with respect to \(\mathbb{P}\) and for any random variable \(X\),
\[\mathbb{E}_{\mathbb{Q}}[X]=\mathbb{E}_{\mathbb{P}}[\Gamma_{t}X]\mbox{ and }\mathbb{E}_{\mathbb{Q}}[X|{\cal F}_{s}]=
\mathbb{E}_{\mathbb{P}}\left .\left [\frac{\Gamma_{t}}{\Gamma_{s}}\cdot X\right |{\cal F}_{s}\right ].\]
If \(X\) is \({\cal F}_{s}\)-measurable, then for \(u\leq s\)
\begin{equation}{\label{kleeq1014}}
\mathbb{E}_{\mathbb{Q}}[X|{\cal F}_{u}]=\mathbb{E}_{\mathbb{P}}\left .\left [\frac{\Gamma_{s}}{\Gamma_{u}}\cdot X\right |{\cal F}_{u}\right ]. \sharp
\end{equation}
It follows immediately from (\ref{kleeq1014})
\begin{equation}{\label{klec1061}}\tag{2}\mbox{}\end{equation}
Corollary \ref{klec1061}. A process \(\{X_{s}\}_{0\leq s\leq t}\) is a \(Q\)-martingale if and only if \(\{\Gamma_{s}X_{s}\}_{0\leq s\leq t}\) is a \(P\)-martingale.
Proof. We see \(X_{s}\) is \({\cal F}_{s}\)-measurable. Then by (\ref{kleeq1014}),
we have
\[X_{u}=\mathbb{E}_{\mathbb{Q}}[X_{s}|{\cal F}_{u}]=\mathbb{E}_{\mathbb{P}}\left .\left [\frac{\Gamma_{s}X_{s}}{\Gamma_{u}}\right |{\cal F}_{u}\right ]=
\frac{\Gamma_{u}X_{u}}{\Gamma_{u}}=X_{u}.\]
This completes the proof. \(\blacksquare\)
Example. (Chung and Williams \cite{chu}) (Brownian motion plus a bounded drift) Suppose \(W\) is a one-dimensional Brownian motion on \((\Omega ,{\cal F},\mathbb{P})\) and \(b:\mathbb{R}\rightarrow \mathbb{R}\) is a bounded measurable function. Then, since \(b\) is bounded, \(\int_{0}^{t}b(W_{s})dW_{s}\) defines a continuous \(L^{2}\)-martingale, and then
\[\Gamma_{t}=\exp\left (\int_{0}^{t}b(W_{s})dW_{s}-\frac{1}{2}\int_{0}^{t}b^{2}(W_{s})ds\right )\]
defines a positive continuous \(L^{2}\)-martingale on \((\Omega ,{\cal F},\mathbb{P})\). For each \(t\geq 0\), let \(P_{t}\) denote the restriction of \(\mathbb{P}\) to \({\cal F}_{t}\) and let \(\mathbb{Q}_{t}\) be the probability measure on \({\cal F}_{t}\) that is absolutely continuous with respect to \(\mathbb{P}_{t}\) and whose Radon-Nikodym derivative is given by
\begin{equation}{\label{chueq927}}
\frac{d\mathbb{Q}_{t}}{d\mathbb{P}_{t}}=\Gamma_{t}\mbox{ on }{\cal F}_{t}.
\end{equation}
Since \(\Gamma_{t}>0\) \(\mathbb{P}\)-a.s., \(\mathbb{Q}_{t}\) is equivalent to \(\mathbb{P}_{t}\), and by the martingale property of \(\{\Gamma_{t}\}_{t\geq 0}\) with respect to \(\{{\cal F}_{t}\}_{t\geq 0}\), the \(\mathbb{Q}_{t}\)’s are consistent, i.e., if \(0\leq s<t\) and \(F\in {\cal F}_{s}\), then
\[\mathbb{Q}_{t}(F)=\mathbb{E}_{\mathbb{P}_{t}}[\Gamma_{t}\cdot 1_{F}]=\mathbb{E}_{\mathbb{P}}[\Gamma_{t}\cdot 1_{F}]=
\mathbb{E}_{\mathbb{P}}[\Gamma_{s}\cdot 1_{F}]=\mathbb{E}_{P_{s}}[\Gamma_{s}\cdot 1_{F}]=\mathbb{Q}_{s}(F).\]
Let
\begin{equation}{\label{chueq928}}
\widehat{W}_{s}=W_{s}-\int_{0}^{s}b(W_{u})du\mbox{ for each }s\geq 0.
\end{equation}
It can be shown that for each \(t\geq 0\), \(\{\widehat{W}_{s}\}_{0\leq s\leq t}\) is a Brownian motion on the interval \([0,t]\) under \(\mathbb{Q}_{t}\). If \(\{(\Gamma_{t},{\cal F}_{t})\}_{t\geq 0}\) is uniformly integrable, the \(\mathbb{Q}_{t}\)’s can be extended to a measure \(\mathbb{Q}\) that is absolutely continuous with respect to \(\mathbb{P}\) on \({\cal F}\). However, in general the uniform integrability does not hold, and there is no such extension. There is a similar change of measure transformation for \(n\)-dimensional Brownian motion. This can be used to represent solutions of some elliptic partial differential equations. It is specified as follows. Suppose \({\bf W}=(W^{(1)},\cdots ,W^{(n)})\) is a \(n\)-dimensional Brownian motion on \((\Omega ,{\cal F},P)\) and \({\bf b}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\) is a bounded Borel measurable function. For each \(t\geq 0\), define
\[\Gamma_{t}=\exp\left (\int_{0}^{t}{\bf b}({\bf W}_{s})d{\bf W}_{s}-\frac{1}{2}\int_{0}^{t}\parallel {\bf b}({\bf X}_{s})\parallel^{2}ds\right ),\]
where
\[\int_{0}^{t}{\bf b}({\bf W}_{s})d{\bf W}_{s}=\sum_{i=1}^{n}\int_{0}^{t}b_{i}({\bf W}_{s})dW_{s}^{(i)}.\]
Using this \(\Gamma_{t}\), define \(Q_{t}\) to be absolutely continuous with respect to \(\mathbb{P}_{t}\) on \({\cal F}_{t}\) so that (\ref{chueq927}) holds. Let \(\{\widehat{{\bf W}}_{t}\}_{t\geq 0}\) be the \(n\)-dimensional process defined as in (\ref{chueq928}). Then on the probability space \((\Omega ,{\cal F}_{t},\mathbb{Q}_{t})\), \(\{\widehat{{\bf W}}_{s}\}_{0\leq s\leq t}\) is a \(n\)-dimensional Brownian motion on the time interval \([0,t]\). \(\sharp\)
Theorem. (Klebaner \cite{kle})(Girsanov’s Theorem for Brownian Motion). Let \(\{W_{s}\}_{0\leq s\leq t}\) be a Brownain motion under probability measure \(\mathbb{P}\), and \(\mu\neq 0\). Consider the process \(\widehat{W}_{s}=W_{s}+\mu s\). There exists a measure \(\mathbb{Q}\) equivalent to \(\mathbb{P}\) such that \(\{\widehat{W}_{s}\}_{0\leq s\leq t}\) is \(\mathbb{Q}\)-Brownian motion. The likelihood is given by
\[\Gamma =\frac{d\mathbb{Q}}{d\mathbb{P}}=\exp\left (-\mu W_{t}-\frac{1}{2}\mu^{2}t\right )\mbox{ and }\frac{d\mathbb{P}}{d\mathbb{Q}}=\frac{1}{\Gamma}=\exp\left (\mu\widehat{W}_{t}-
\frac{1}{2}\mu^{2}t\right ).\]
Proof. The proof uses Levy’s characterization of Brownian motion, as a continuous martingale with quadratic variation process \(t\). Quadratic variation is a path property, it does not depend on the probability measure, as long as it is absolutely continuous with respect to the given one. Therefore using the fact that \(\mu s\) is smooth and does not contribute to the quadratic variation
\[\langle\widehat{W}\rangle_{s}=\langle W\rangle_{s}=s.\]
It remains to establish that \(\widehat{W}\) is a \(\mathbb{Q}\)-martingale. Let \(\Gamma_{s}=\mathbb{E}_{\mathbb{P}}[\Gamma |{\cal F}_{s}]\). Then \(\{\Gamma_{s}\}_{0\leq s\leq t}\) is a \(\mathbb{P}\)-martingale. By Corollary \ref{klec1061}, it is enough to show that \(\{\Gamma_{s}\widehat{W}_{s}\}_{0\leq s\leq t}\) is a \(\mathbb{P}\)-martingale,
\[\mathbb{E}_{\mathbb{P}}\left .\left [\widehat{W}_{s}\Gamma_{s}\right |{\cal F}_{u}\right ]=
\mathbb{E}_{\mathbb{P}}\left .\left [(W_{s}+\mu s)\cdot\exp\left (-\mu W_{s}-\frac{1}{2}\mu^{2} s\right )\right |{\cal F}_{u}\right ]=\widehat{W}_{u}\Gamma_{u}.\]
This is done by direct calculations. \(\blacksquare\)
It turns out that not only linear drift can be removed by a chainge of measure, but any drift of the form \(\int_{0}^{t}H_{s}ds\) with a predictable process \(H\).
Theorem. (Klebaner \cite{kle})(Girsanov’s Theorem for Removal of Drift). Let \(\{W_{s}\}_{0\leq s\leq t}\) be a \(\mathbb{P}\)-Brownian motion, and \(\{H_{s}\}_{0\leq s\leq t}\) be a predictable process with \(\int_{0}^{t} H_{s}^{2}ds<\infty\). Let \(X_{s}=-\int_{0}^{s}H_{u}dW_{u}\) and assume that the stochastic exponential \({\cal E}(X)\) is a martingale (a sufficient condition for this is \(\mathbb{E}\left [\exp\left (\frac{1}{2}\int_{0}^{t}H_{s}^{2} ds\right )\right ]<\infty\) by the Novikov’s condition in Proposition \ref{klet811}). Then there is a measure \(Q\) equivalent to \(\mathbb{P}\) such that the process
\[\widehat{W}_{s}=W_{s}+\int_{0}^{s}H_{u}du\]
is a \(\mathbb{Q}\)-Brownian motion. The likelihood is given by
\[\Gamma =\frac{d\mathbb{Q}}{d\mathbb{P}}=\exp\left (-\int_{0}^{t}H_{s}dW_{s}-\frac{1}{2}\int_{0}^{t}H_{s}^{2}ds\right )={\cal E}(X).\]
Proof. The proof is similar to the previous one and we only sketch it. Notice that quadratic variation of \(\widehat{W}_{s}=W_{s}+\int_{0}^{s}H_{u}du\) is \(s\), since the integral is continuous and is of finite variation. \(\widehat{W}\) is clearly continuous. We indicate how to establish that \(W\) is a \(\mathbb{Q}\)-martingale. By Corollary \ref{klec1061}, it is enough to show that \(\{\Gamma_{s}\widehat{W}_{s}\}_{0\leq s\leq t}\) is a \(\mathbb{P}\)-martingale with \(\Gamma_{s}=\mathbb{E}_{\mathbb{P}}[\Gamma |{\cal F}_{s}]\). Consider now
\[\Gamma_{s}\cdot {\cal E}(\widehat{W})_{s}={\cal E}(X)_{s}\cdot {\cal E}(\widehat{W})_{s}={\cal E}(X+\widehat{W}+\langle X,\widehat{W}\rangle )_{s}={\cal E}(X+W)_{s},\]
where we have used the rule for the product of stochastic exponential in Proposition \ref{klet89}, and that \(\langle X,\widehat{W}\rangle_{s}=-\int_{0}^{s}H_{u}du\) and \(\widehat{W}_{s}=W_{s}+\langle X,\widehat{W}\rangle_{s}\). Since \(X\) and \(W\) are both \(P\)-martingales, so is \(X+W\). Consequently \({\cal E}(X+W)\) is also a \(\mathbb{P}\)-martingale. Hence \(\{\Gamma_{s}\cdot {\cal E}(\widehat{W})_{s}\}_{0\leq s\leq t}\) is a \(\mathbb{P}\)-martingale, and consequently \(\{{\cal E}(\widehat{W})_{s}\}_{0\leq s\leq t}\) is a \(\mathbb{Q}\)-martingale.
This implies that \(\{\widehat{W}_{s}\}_{0\leq s\leq t}\) is also a \(\mathbb{Q}\)-martingale. \(\Gamma ={\cal E}(X)\) follows from Proposition \ref{klet88}. \(\blacksquare\)
The same result holds for \(n\)-dimensional Brownian motions. If \({\bf W}\) is an \(n\)-dimensional Brownian motion and \({\bf H}\) is a predictable process, then provided \(\mathbb{E}\left [\exp\left (\frac{1}{2}\int_{0}^{t}\parallel {\bf H}_{s}\parallel^{2}ds\right )\right ]<\infty\), \(\widehat{{\bf W}}=(\widehat{W}_{s}^{(1)},\cdots ,\widehat{W}_{s}^{(n)})\), where
\[\widehat{W}_{s}^{(i)}=W_{s}^{(i)}+\int_{0}^{s}H_{u}^{(i)}du,\]
is \(\mathbb{Q}\)-Brownian motion. Here \(d\mathbb{Q}={\cal E}(X)_{t}d\mathbb{P}\) with
\[X_{s}=-\sum_{i=1}^{n}\int_{0}^{s}H_{u}^{(i)}dW_{u}^{(i)}=-({\bf H}\cdot {\bf W})_{s}.\]
The following discussion follows from Karatzas and Shreve \cite{kar}. We begin with a \(n\)-dimensional Brownian motion under \(\mathbb{P}\), and then construct a new measure \(\widehat{\mathbb{P}}\) under which a
“translated” process is a \(n\)-dimensional Brownian motion.
Let \(\{({\bf W}_{t},{\cal F}_{t})\}_{t\geq 0}\) be a \(n\)-dimensional Brownian motion defined on a probability space \((\Omega ,{\cal F},\mathbb{P})\) with \(\mathbb{P}\{{\bf W}_{0}={\bf 0}\}=1\). We assume that the filtration \(\{{\cal F}_{t}\}_{t\geq 0}\) satisfies the usual conditions. Let \(\{({\bf X}_{t}=(X_{t}^{(1)},\cdots ,X_{t}^{(n)}),{\cal F}_{t})\}_{t\geq 0}\) be a vector of measurable adapted processes satisfying
\begin{equation}{\label{kareq351}}
P\left\{\int_{0}^{t}(X_{s}^{(i)})^{2}ds<\infty\right\}=1\mbox{ for }1\leq i\leq n,t\geq 0.
\end{equation}
Then, for each \(i\), the stochastic integral \(\int_{0}^{t}X_{s}^{(i)}dW_{s}^{(i)}\) is defined and is a continuous local martingale. We set
\begin{equation}{\label{kareq352}}
Z_{t}({\bf X})\equiv\exp\left (\sum_{i=1}^{n}\int_{0}^{t}X_{s}^{(i)}dW_{s}^{(i)}-\frac{1}{2}\int_{0}^{t}\parallel {\bf X}_{s}\parallel^{2}ds
\right ).
\end{equation}
Just as in Example \ref{kare339}, \(Z({\bf X})\) satisfies the following equation
\begin{equation}{\label{kareq353}}
Z_{t}({\bf X})=1+\sum_{i=1}^{n}\int_{0}^{t}Z_{s}({\bf X})X_{s}^{(i)}dW_{s}^{(i)},
\end{equation}
which shows that \(Z({\bf X})\) is a continuous local martingale with \(Z_{0}({\bf X})=1\). Under certain conditions on \({\bf X}\), \(Z({\bf X})\) will in fact be a martingale, and so \(E[Z_{t}({\bf X})]=1\) for \(t\geq 0\), which will be discussed later. In this case, we can define, for each \(t\geq 0\), a probability measure \(\widehat{\mathbb{P}}_{t}\) on \({\cal F}_{t}\) by
\[\widehat{\mathbb{P}}_{t}(A)\equiv \mathbb{E}\left[1_{A}\cdot Z_{t}({\bf X})\right ]=\int_{A}Z_{t}({\bf X})d\mathbb{P}\mbox{ for }A\in {\cal F}_{t}.\]
(We see that \(\widehat{\mathbb{P}}(\Omega )=\mathbb{E}\left[1_{\Omega}\cdot Z_{t}({\bf X})\right ]=E[Z_{t}({\bf X})]=1\).) Sometimes we write \(d\widehat{\mathbb{P}}_{t}=Z_{t}({\bf X})d\mathbb{P}\). The martingale property shows that the family of probability measures \(\{\widehat{\mathbb{P}}_{t}\}_{t\geq 0}\) satisfies the consistency condition
\[\widehat{\mathbb{P}}_{t}(A)=\widehat{\mathbb{P}}_{s}(A)\mbox{ for }A\in {\cal F}_{s},0\leq s\leq t.\]
\begin{equation}{\label{kart351}}\tag{3}\mbox{}\end{equation}
Theorem \ref{kart351}. (Karatzas and Shreve \cite{kar})(Girsanov’s Theorem). Assume that \(Z({\bf X})\) defined by \((\ref{kareq352})\) is a martingale. Define a process \(\{(\widehat{{\bf W}}_{t}=(\widehat{W}_{t}^{(1)},\cdots ,\widehat{W}_{t}^{(n)}),{\cal F}_{t})\}_{t\geq 0}\) by
\begin{equation}{\label{kareq356}}
\widehat{W}_{t}^{(i)}\equiv W_{t}^{(i)}-\int_{0}^{t}X_{s}^{(i)}ds\mbox{ for }1\leq i\leq n,t\geq 0.
\end{equation}
For each fixed \(t\geq 0\), the process \(\{(\widehat{W}_{s},{\cal F}_{s})\}_{0\leq s\leq t}\) is a \(n\)-dimensional Brownian motion on \((\Omega ,{\cal F}_{t},\widehat{\mathbb{P}}_{t})\). \(\sharp\)
Occasionally, one wants \(\widehat{{\bf W}}\), as a process defined for all \(t\in [0,\infty )\), to be a Brownian motion, and for this purpose the measures \(\{\widehat{\mathbb{P}}_{t}:0\leq t<\infty\}\) are inadequate. We
would like to have a single measure \(\widehat{\mathbb{P}}\) defined on \({\cal F}_{\infty}\), so that \(\widehat{\mathbb{P}}\) restricted to any \({\cal F}_{t}\) agrees with \(\widehat{\mathbb{P}}_{t}\); however, such a measure does not exist in general. We thus content ourselves with a measure \(\widehat{\mathbb{P}}\) defined only on \({\cal F}_{\infty}^{\bf W}\), the \(\sigma\)-field generated by \({\bf W}\), such that \(\widehat{\mathbb{P}}\) restricted to any \({\cal F}_{t}^{\bf W}\) agrees with \(\widehat{\mathbb{P}}_{t}\), i.e.
\[\widehat{\mathbb{P}}(A)=\mathbb{E}[1_{A}\cdot Z_{t}({\bf X})]\mbox{ for }A\in {\cal F}_{t}^{\bf W}, t\geq 0.\]
If such a \(\widehat{\mathbb{P}}\) exists, it is clearly unique. The existence of \(\widehat{\mathbb{P}}\) follows from the Daniell-Kolmogorov consistency theorem. The process \(\widehat{{\bf W}}\) in Theorem \ref{kart351} is adapted to the filtration \(\{{\cal F}_{t}\}_{t\geq 0}\), and so is the process \(\left\{\int_{0}^{t}X_{s}^{(i)}ds\right\}_{t\geq 0}\). However, when working with the measure \(\widehat{\mathbb{P}}\) which is defined only on \({\cal F}_{\infty}^{{\bf W}}\), we wish \(\widehat{{\bf W}}\) to be adapted to \(\{{\cal F}_{t}^{\bf W}\}_{t\geq 0}\). This filtration does not satisfy the usual conditions, and so we must impose the stronger condition of progressive measurability on \({\bf X}\). We have the following corollary.
\begin{equation}{\label{{karc352}}\tag{4}\mbox{}\end{equation}
Corollary \ref{karc352}. Let \(\{({\bf W}_{t},{\cal F}_{t})\}_{t\geq 0}\) be a \(n\)-dimensional Brownian motion on \((\Omega ,{\cal F},P)\) with \(P\left\{{\bf W}_{0}={\bf 0}\right\}=1\), and assume that the filtration \(\{{\cal F}_{t}\}_{t\geq 0}\) satisfies the usual conditions. Let \(\{({\bf X}_{t},{\cal F}_{t}^{\bf W})\}_{t\geq 0}\) be a \(n\)-dimensional progressively measurable process satisfying \((\ref{kareq351})\). If \(Z({\bf X})\) of \((\ref{kareq352})\) is a martingale, then \(\{(\widehat{{\bf W}}_{t},{\cal F}_{t}^{\bf W})\}_{t\geq 0}\) defined by \((\ref{kareq356})\) is a \(n\)-dimensional Brownian motion on \((\Omega ,{\cal F}_{\infty}^{\bf W},\widehat{\mathbb{P}})\).
Proof. For \(0\leq t_{1}<\cdots <t_{k}\leq t\), we have
\[\widehat{\mathbb{P}}\left\{(\widehat{W}_{t_{1}},\cdots ,\widehat{W}_{t_{k}})\in A\right\}=\widehat{\mathbb{P}}_{t}\left\{(\widehat{W}_{t_{1}},\cdots ,
\widehat{W}_{t_{k}})\in A\right\}\mbox{ for }A\in {\cal B}(\mathbb{R}^{kn}).\]
The result now follows from Theorem \ref{kart351}. \(\blacksquare\)
Under the assumptions of Corollary \ref{karc352}, the probability measure \(\mathbb{P}\) and \(\widehat{\mathbb{P}}\) are mutually absolutely continuous when restricted to \({\cal F}_{t}^{\bf W}\) for \(t\geq 0\). However, viewed as probability measures on \({\cal F}_{\infty}^{\bf W}\), \(\mathbb{P}\) and \(\widehat{\mathbb{P}}\) may not be mutually absolutely continuous.
In order to use the Girsanov theorem effectively, we need some fairly general conditions under which the process \(Z({\bf X})\) defined by (\ref{kareq352}) is a martingale. This process is a local martingale because
of (\ref{kareq353}). Indeed, with
\[T_{m}=\inf\left\{t\geq 0:\max_{1\leq i\leq n}\int_{0}^{t}(Z_{s}({\bf X})\cdot X_{s}^{(i)})^{2}ds=m\right\},\]
the “stopped” processes \(\{(Z_{t\wedge T_{m}}({\bf X}),{\cal F}_{t})\}_{t\geq 0}\) are martingales. Consequently, we have
\[\mathbb{E}\left.\left [Z_{t\wedge T_{m}}\right |{\cal F}_{s}\right ]=Z_{s\wedge T_{m}}\mbox{ for }0\leq s\leq t,\]
and using Fatou’s lemma as \(m\rightarrow\infty\), we obtain \(\mathbb{E}\left.\left [Z_{t}({\bf X})\right |{\cal F}_{s}\right ]\leq Z_{s}({\bf X})\) for \(0\leq s\leq t\). In other words, \(Z({\bf X})\) is always
a supermartingale and is a martingale if and only if
\begin{equation}{\label{kareq3517}}
\mathbb{E}\left[Z_{t}({\bf X})\right ]=\mathbb{E}\left[Z_{0}({\bf X})\right ]=1\mbox{ for }t\geq 0
\end{equation}
from Proposition \ref{klet73}. We provide now sufficient conditions for (\ref{kareq3517}).
Proposition. Let \(\{(M_{t},{\cal F}_{t})\}_{t\geq 0}\) be a continuous local martingale and define \(Z_{t}=\exp\left (M_{t}-\frac{1}{2}\langle M\rangle_{t}\right )\) for \(t\geq 0\). If \(\mathbb{E}\left[\exp\left (\frac{1}{2}\langle M\rangle_{t}\right )\right ]<\infty\) for \(t\geq 0\), then \(\mathb{E}[Z_{t}]=1\) for \(t\geq 0\). \(\sharp\)
\begin{equation}{\label{karc3513}}\tag{5}\mbox{}\end{equation}
Corollary \ref{karc3513}. (Novikov Condition). Let \(\{({\bf W}_{t}=(W_{t}^{(1)},\cdots ,W_{t}^{(n)}),{\cal F}_{t})\}_{t\geq 0}\) be a \(n\)-dimensional Brownian motion, and let \(\{({\bf X}_{t}=(X_{t}^{(1)},\cdots ,X_{t}^{(n)}),{\cal F}_{t})\}_{t\geq 0}\) be a vector of measurable adapted processes satisfying (\ref{kareq351}). If
\begin{equation}{\label{kareq3520}}
\mathbb{E}\left[\exp\left (\frac{1}{2}\int_{0}^{t}\parallel X_{s}\parallel^{2}ds\right )\right ]<\infty\mbox{ for }t\geq 0,
\end{equation}
then \(Z({\bf X})\) defined by \((\ref{kareq352})\) is a martingale. \(\sharp\)
Corollary (Novikov Condition). Corollary \ref{karc3513} still holds if (\ref{kareq3520}) is replaced by the following assumption: there exists a sequence \(\{t_{n}\}_{n\in {\bf N}}\) of real numbers with \(0=t_{0}<t_{2}<\cdots <t_{n}\uparrow\infty\), such that
\[\mathbb{E}\left[\exp\left (\frac{1}{2}\int_{t_{n-1}}^{t_{n}}\parallel {\bf X}_{s}\parallel^{2}ds\right )\right ]<\infty. \sharp\]
Definition. Let \(C^{n}([0,\infty ))\) be the space of continuous functions \({\bf x}:[0,\infty )\rightarrow \mathbb{R}^{n}\). For \(t\geq 0\), define \({\cal G}_{t}=\sigma ({\bf x}(s):0\leq s\leq t)\), and set $latex {\cal
G}={\cal G}_{\infty}$. A progressively measurable functional on \(C^{n}([0,\infty ))\) is a mapping \(\mu :[0,\infty )\times C^{n}([0,\infty ))\rightarrow \mathbb{R}\) which has the property that for each fixed \(t\geq 0\), \(\mu\) restricted to \([0,t]\times C^{n}([0,\infty ))\) is \(({\cal B}([0,t])\times {\cal G}_{t})\times {\cal B}(\mathbb{R})\)-measurable. \(\sharp\)
If \((\mu_{1},\cdots ,\mu_{n})\) is a vector of progressively measurable functionals on \(C^{n}([0,\infty ))\) and \(\{({\bf W}_{t}=(W_{t}^{(1)},\cdots ,W_{t}^{(n)}),{\cal F}_{t})\}_{t\geq 0}\) is a \(n\)-dimensional Brownian motion on some spaces \((\Omega ,{\cal F},P)\), then the processes
\begin{equation}{\label{kareq3522}}
X_{t}^{(i)}(\omega )\equiv\mu_{i}(t,{\bf W}_{\cdot}(\omega ))\mbox{ for }1\leq i\leq n,t\geq 0
\end{equation}
are progressively measurable relative to \(\{{\cal F}_{t}\}_{t\geq 0}\).
Corollary. Let the vector \(\mbox{\boldmath \)latex \mu$}=(\mu_{1},\cdots ,\mu_{n})$ of progressively measurable functionals on \(C^{n}([0,\infty ))\) satisfy, for each \(t\geq 0\) and some \(c_{t}>0\) depending on \(t\), the condition
\[|\mu (t,{\bf x})|\leq c_{t}\cdot (1+{\bf x}^{*}(t))\mbox{ for }t\geq 0,\]
where \({\bf x}^{*}(t)\equiv\max_{0\leq s\leq t}\parallel {\bf x}(s)\parallel\). Then with \({\bf X}_{t}=(X_{t}^{(1)},\cdots ,X_{t}^{(n)})\) defined by \((\ref{kareq3522})\), \(Z({\bf X})\) of \((\ref{kareq352})\) is a martingale. \(\sharp\)
The following discussion follows from Friedman \cite{fri}.
\begin{equation}{\label{frit711}}\tag{6}\mbox{}\end{equation}
Proposition \ref{frit711}. (Friedman \cite{fri}). Let \({\bf X}\in {\cal L}^{2}([0,\eta ])\) and assume that there exists positive numbers \(h,k\) such that
\[\mathbb{E}\left[\exp\left (h\cdot\parallel X_{t}\parallel^{2}\right )\right ]\leq k\mbox{ for }0\leq t\leq\eta .\]
Then
\[\mathbb{E}\left[\exp\left (\int_{t_{1}}^{t_{2}}{\bf X}_{s}d{\bf W}_{s}-\frac{1}{2}\int_{t_{1}}^{t_{2}}\parallel X_{s}\parallel^{2}ds\right )\right ]=1\mbox{ if }0\leq t_{1}<t_{2}\leq\eta . \sharp\]
Corollary. Under the assumptions of Proposition \ref{frit711}, then we have
\[\mathbb{E}\left.\left [\exp\left (\int_{t_{1}}^{t_{2}}{\bf X}_{s}d{\bf W}_{s}-\frac{1}{2}\int_{t_{1}}^{t_{2}}\parallel X_{s}\parallel^{2}ds\right )\right |{\cal F}_{t_{1}}\right ]=1\mbox{ a.s.}\]
for any \(0\leq t_{1}<t_{2}\leq\eta\). \(\sharp\)
\begin{equation}{\label{fric715}}\tag{7}\mbox{}\end{equation}
Corollary \ref{fric715}. For any \(X\in {\cal L}^{2}([0,\eta ])\), we have
\[\mathbb{E}\left[\exp\left (\int_{t_{1}}^{t_{2}}{\bf X}_{s}d{\bf W}_{s}-\frac{1}{2}\int_{t_{1}}^{t_{2}}\parallel X_{s}\parallel^{2}ds\right )\right ]\leq 1\]
and
\[\mathbb{E}\left.\left [\exp\left (\int_{t_{1}}^{t_{2}}{\bf X}_{s}d{\bf W}_{s}-\frac{1}{2}\int_{t_{1}}^{t_{2}}\parallel X_{s}\parallel^{2}ds\right )\right |{\cal F}_{t_{1}}\right ]\leq 1\mbox{ a.s.}\]
for any \(0\leq t_{1}<t_{2}\leq\eta\). \(\sharp\)
Theorem. (Girsanov’s Theorem). Let \(\{{\bf W}_{t}\}_{0\leq t\leq\eta}\) be an \(n\)-dimensional Brownian motion and let \({\bf X}\in {\cal L}^{2}([0,\eta ])\). Define
\[\Gamma =\exp\left (\int_{0}^{\eta}{\bf X}_{t}d{\bf W}_{t}-\frac{1}{2}\int_{0}^{\eta}\parallel X_{t}\parallel^{2}dt\right ),\]
\[\widehat{{\bf W}}_{t}={\bf W}_{t}-\int_{0}^{t}{\bf X}_{s}ds,\]
\[d\widehat{\mathbb{P}}=\Gamma d\mathbb{P}.\]
If
\begin{equation}{\label{frieq724}}
\widehat{\mathbb{P}}(\Omega )=1,
\end{equation}
then \(\{\widehat{{\bf W}}_{t}\}_{0\leq t\leq\eta}\) is an \(n\)-dimenssional \(\widehat{\mathbb{P}}\)-Brownian motion. \(\sharp\)
We have established sufficient conditions for (\ref{frieq724}) to hold in Proposition \ref{frit711}. We als have \(\widehat{\mathbb{P}}(\Omega )\leq 1\) for any \({\bf X}\in {\cal L}^{2}([0,\eta ])\) in Corollary \ref{fric715}.
Definition. A process \(\{{\bf X}_{t}\}_{0\leq t\leq\eta}\) is called an Ito process with respect to \(({\bf W}_{t},P,{\cal F}_{t})\) relative to the pair \((\mbox{\boldmath \)latex \mu$}_{t},\mbox{\boldmath \(\sigma\)}_{t})$ if
\[{\bf X}_{t}={\bf X}_{0}+\int_{0}^{t}\mbox{\boldmath \(\mu\)}_{s}ds+\int_{0}^{t}\mbox{\boldmath \(\sigma\)}_{s}d{\bf W}_{s}\mbox{ for }0\leq t\leq\eta ,\]
where \(\mbox{\boldmath \)latex \mu$}$ is an \(n\)-vector in \({\cal L}^{1}([0,\eta ])\) and \(\mbox{\boldmath \)latex \sigma$}$ is an \(n\times n\) matrix in \({\cal L}^{2}([0,\eta ])\). \(\sharp\)
Theorem. (Girsanov’s Theorem). Let \(\{{\bf X}_{t}\}_{0\leq t\leq\eta}\) be an Ito process with respect to \(({\bf W}_{t},P,{\cal F}_{t})\) relative to the pair \((\mbox{\boldmath \)latex \mu$}_{t},\mbox{\boldmath \(\sigma\)}_{t})$. Let \({\bf X}\in {\cal L}^{2}([0,\eta ])\) and set
\[\Gamma =\exp\left (\int_{0}^{\eta}{\bf X}_{t}d{\bf W}_{t}-\frac{1}{2}\int_{0}^{\eta}\parallel X_{t}\parallel^{2}dt\right ),\]
\[\widehat{{\bf W}}_{t}={\bf W}_{t}-\int_{0}^{t}{\bf X}_{s}ds,\]
\[d\widehat{\mathbb{P}}=\Gamma d\mathbb{P}.\]
Assume that \(\widehat{\mathbb{P}}(\Omega )=1\). Then \(\{\widehat{{\bf W}}_{t}\}_{0\leq t\leq\eta}\) is an \(n\)-dimenssional \(\widehat{\mathbb{P}}\)-Brownian motion and the process \(\{{\bf Y}_{t}\}_{0\leq t\leq\eta}\) is an Ito process with respect to \((\widehat{{\bf W}}_{t},\widehat{\mathbb{P}},{\cal F}_{t})\) relative to the pair \((\widehat{\mbox{\boldmath \)latex \mu$}}_{t},\mbox{\boldmath \(\sigma\)}_{t})$, where \(\widehat{\mbox{\boldmath \)latex \mu$}}_{t}=\mbox{\boldmath \(\mu\)}_{t}+\mbox{\boldmath \(\sigma\)}_{t}\cdot {\bf X}_{t}$. \(\sharp\)
The following discussion follows from Chung and Williams \cite{chu}. We now study the behavior of local martingales under mutually absolutely continuous changes of probability measure. As usual, \((\Omega ,{\cal F},P)\) is a complete probability space with a given standard filtration \(\{{\cal F}_{t}\}_{t\geq 0}\) and \({\cal F}_{\infty}=\bigve\mathbb{E}_{t\geq 0} {\cal F}_{t}\).
We suppose \(Q\) is a probability measure on \((\Omega ,{\cal F})\) such that \(Q\) is equivalent to \(P\). Let \(\Gamma =d\mathbb{Q}/d\mathbb{P}\), the Radon-Nikodym derivative of \(Q\) with respect to \(P\). For each \(t\geq 0\), define
\[\Gamma_{t}=\mathbb{E}_{\mathbb{P}}[\Gamma |{\cal F}_{t}].\]
Then \(\{(\Gamma_{t},{\cal F}_{t})\}_{t\geq 0}\) is a uniformly integrable martingale on \((\Omega ,{\cal F},P)\). We may suppose it is right-continuous, since there is always a right-continuous version of it.
Moreover, \(\Gamma_{\infty}=E[\Gamma |{\cal F}_{\infty}]\) is \(P\)-a.s. equal to \(\lim_{t\rightarrow\infty}\Gamma_{t}\) by the martingale convergence theorem.
Proposition. (Chung and Williams \cite{chu}). Let \(\{M_{t}\}_{t\geq 0}\) be a right-continuous stochastic process on \((\Omega ,{\cal F})\) with \(M_{t}\in {\cal F}_{t}\) for all \(t\geq 0\). Then \(\{(\Gamma_{t}\cdot M_{t},{\cal F}_{t})\}_{t\geq 0}\) is a local martingale under \(P\) if and only if \(\{(M_{t},{\cal F}_{t})\}_{t\geq 0}\) is a local martingale under \(Q\). \(\sharp\)
Theorem. (Chung and Williams \cite{chu})(Girsanov Transformation). Suppose \(\{(X_{t},{\cal F}_{t})\}_{t\geq 0}\) is a continuous local martingale under \(\mathbb{P}\). Let \(\mathbb{Q}\), \(\Gamma\) and \(\{(\Gamma_{t},{\cal F}_{t})\}_{t\geq 0}\) be as before. Suppose the martingale \(\{(\Gamma_{t},{\cal F}_{t})\}_{t\geq 0}\) has a continuous version. Using such a continuous version, let
\[A_{t}=\int_{0}^{t}\frac{1}{\Gamma_{s}}d\langle\Gamma ,X\rangle_{s},\]
wherever the right member is well-defined and finite for all \(t\). Then \(\{A_{t}\}_{t\geq 0}\) is well-defined \(\mathbb{P}\)-a.s. If \(A\) is defined to be identically zero on the remaining \(\mathbb{P}\)-null set, then \(A\) is locally of bounded variation \((\)under either \(\mathbb{P}\) or \(\mathbb{Q})\) and \(\{X_{t}-A_{t}\}_{t\geq 0}\) is a continuous local martingale under \(\mathbb{Q}\), with a quadratic variation process that is the same as that for \(X\) under \(\mathbb{P}\). \(\sharp\)
Thus, under the change of probability from \(\mathbb{P}\) to \(\mathbb{Q}\), \(X\) changes from a continuous local martingale under \(P\) to a continuous semimartingale (the sum of a continuous local martingale and a continuous process that is locally of bounded variation) under \(\mathbb{Q}\).
The following discussion follows from Protter \cite{pro}. Let \(X\) be a semimartingale on a space \((\Omega ,{\cal F},\{{\cal F}_{t}\}_{t\geq 0},P)\) satisfying the usual condition. We see that if \(Q\) is another probability measure on \((\Omega ,{\cal F})\) and \(Q\ll P\), then \(X\) is a \(\mathbb{Q}\)-semimartingale as well. If \(X\) is a classical semimartingale (or equivalently, if \(X\) is decomposable) and has a decomposition \(X=M+A\), \(M\) a local martingale and \(A\) an FV process, then it is often useful to be able to calculate the analogous decomposition \(X=N+B\), if it exists under \(Q\). This is rather tricky unless we make a simplifying assumption which usually holds in practice: that both \(\mathbb{Q}\ll \mathbb{P}\) and \(\mathbb{P}\ll \mathbb{Q}\).
If \(\mathbb{Q}\ll \mathbb{P}\), then there exists a random variable \(Z\) in \(L^{1}(\mathbb{P})\) such that
\[\frac{d\mathbb{Q}}{d\mathbb{P}}=Z\mbox{ and }\mathbb{E}_{\mathbb{P}}[Z]=1.\]
We let
\[Z_{t}=\mathbb{E}_{\mathbb{P}}[Z|{\cal F}_{t}]=\mathbb{E}_{\mathbb{P}}\left [\left .\frac{d\mathbb{Q}}{d\mathbb{P}}\right |{\cal F}_{t}\right ]\]
be the right-continuous version. Then \(\{Z_{t}\}_{t\geq 0}\) is a uniformly integrable martingale and hence a semimartingale by Corollary~\ref{proc314}. Note that if \(\mathbb{Q}\) is equivalent to \(\mathbb{P}\), then
\[\frac{d\mathbb{P}}{d\mathbb{Q}}\in L^{1}(Q)\mbox{ and }\frac{d\mathbb{P}}{d\mathbb{Q}}=\left (\frac{d\mathbb{Q}}{d\mathbb{P}}\right )^{-1}.\]
Proposition. (Protter \cite{pro}). Let \(\mathbb{Q}\) be equivalent to \(\mathbb{P}\) and \(Z_{t}=\mathbb{E}_{\mathbb{P}}[Z|{\cal F}_{t}]=\mathbb{E}_{\mathbb{P}}\left [\left .\frac{d\mathbb{Q}}{d\mathbb{P}}\right |{\cal F}_{t}\right ]\). An adapted, RCLL process \(M\) is a \(\mathbb{Q}\)-local martingale if and only if \(MZ\) is a \(\mathbb{P}\)-local martingale. \(\sharp\)
Theorem. (Protter \cite{pro})(Girsanov-Meyer). Let \(\mathbb{Q}\) be equivalent to \(\mathbb{P}\) and \(Z_{t}=\mathbb{E}_{\mathbb{P}}[Z|{\cal F}_{t}]=\mathbb{E}_{\mathbb{P}}\left [\left .\frac{d\mathbb{Q}}{d\mathbb{P}}\right |{\cal F}_{t}\right ]\). Let \(X\) be a classical semimartingale under \(\mathbb{P}\) with decomposition \(X=M+A\). Then \(X\) is also a classical semimartingale
under \(\mathbb{Q}\) and has a decomposition \(X=L+C\), where
\[L_{t}=M_{t}-\int_{0}^{t}\frac{1}{Z_{s}}d\langle Z,M\rangle _{s}\]
is a \(\mathbb{Q}\)-local martingale, and \(C=X-L\) is a \(\mathbb{Q}\)-FV process. \(\sharp\)
\begin{equation}{\label{prot321}}\tag{8}\mbox{}\end{equation}
Theorem \ref{prot321}. {\em (Protter \cite{pro})} Let \(W\) be a standard Brownian motion on \((\Omega ,{\cal F},\{{\cal F}_{t}\}_{t\geq 0},P)\), and let \(H\in {\bf L}\) be bounded. Let
\[\widehat{W}_{s}=\int_{0}^{s}H_{u}du+W_{s}\]
and define \(\mathbb{Q}\) by
\[\frac{d\mathbb{Q}}{d\mathbb{P}}=\exp\left (-\int_{0}^{t}H_{u}dW_{u}-\frac{1}{2}\int_{0}^{t}H_{u}^{2}du\right ).\]
for some \(t>0\). Then under \(\mathbb{Q}\), \(\widehat{W}\) is a standard Brownian motion for \(0\leq s\leq t\).
Proof. Let
\[Z_{s}=\exp\left (-\int_{0}^{s}H_{u}dW_{u}-\frac{1}{2}\int_{0}^{s}H_{u}^{2}du\right ).\]
Then if \(Z_{s}=E[Z_{t}|{\cal F}_{s}]\), we know by Proposition \ref{prot236} that \(Z\) satisfies the equation
\[Z_{s}=1-\int_{0}^{s}Z_{u-}H_{u}dW_{u}.\]
By the Girsanov-Meyer Theorem, we know that
\[N_{s}=W_{s}-\int_{0}^{s}\frac{1}{Z_{u}}d\langle Z,W\rangle _{u}\]
is a \(\mathbb{Q}\)-local martingale. However
\[\langle Z,W\rangle _{s}=\langle -ZH\bullet W,W\rangle _{s}=\int_{0}^{s}-Z_{u}H_{u}d\langle W\rangle _{u}=-\int_{0}^{s}Z_{u}H_{u}du,\]
since \(\langle W\rangle _{s}=s\) for Brownian motion. Therefore
\[N_{s}=W_{s}-\int_{0}^{s}-\frac{1}{Z_{u}}Z_{u}H_{u}du=W_{s}+\int_{0}^{s}H_{u}du=\widehat{W}_{s},\]
and therefore \(\widehat{W}\) is a \(Q\)-local martingale. Since \(\{\int_{0}^{s}H_{u}du\}_{t\geq 0}\) is a continuous FV process we have that \(\langle\widehat{W}\rangle _{s}=\langle W\rangle _{s}=s\), and therefore by L\'{e}vy’s theorem in Proposition \ref{prot238}, we conclude that \(\widehat{W}\) is a standard Brownian motion. \(\blacksquare\)
Corollary. Let \(W\) be a standard Brownian motion and \(H\in {\bf L}\) be bounded. Then
\[\widehat{W}_{s}=\int_{0}^{s}H_{u}du+W_{s}\]
for \(0\leq s\leq t<\infty\) is equivalent to Wiener measure.
Proof. Let \(C[0,t]\) be the space of continuous functions on \([0,t]\) with values in \(\mathbb{R}\) (such a space is called a path space). If \(W=\{W_{s}\}_{s\geq 0}\) is a standard Brownian motion, it induces a
measure \(\mu_{W}\) on \(C[0,t]\) by
\[\mu_{W}(A)=P\left\{\omega :s\mapsto W_{s}(\omega )\in A\right\}.\]
Let \(\mu_{\widehat{W}}\) be the analogous measure induced by \(\widehat{W}\). Then by Theorem \ref{prot321}, we have that \(\mu_{\widehat{W}}\) is equivalent to \(\mu_{W}\) and further we have
\[\frac{d\mu_{W}}{d\mu_{\widehat{W}}}=\exp\left (-\int_{0}^{t}H_{u}dW_{u}-\frac{1}{2}\int_{0}^{t}H_{u}^{2}du\right ).\]
This completes the proof. \(\blacksquare\)
To indicate another use of the Girsanov-Meyer theorem let us consider stochastic differential equations. Let \(W\) be a standard Brownian motion on a space \((\Omega ,{\cal F},\{{\cal F}_{t}\}_{t\geq 0},\mathbb{P})\) satisfying the usual condition. Let \(f_{i}(\omega ,s,x)\) be functions, i=1,2, satisfying
- (i) \(|f_{i}(\omega ,s,x)-f_{i}(\omega ,s,y)|\leq c\cdot |x-y|\) for fixed \((\omega ,s)\)
- (ii) \(f_{i}(\cdot ,s,x)\in {\cal F}_{s}\) for fixed \((s,x)\)
- (iii) \(f_{i}(\omega ,\cdot ,x)\) is left-continuous with right limits for fixed \((\omega ,x)\).
By a Picard-type iteration procedure one can show there exists a unique solution (with continuous paths) of
\begin{equation}{\label{proeq5}}
X_{t}=X_{0}+\int_{0}^{t}f_{1}(\cdot ,s,X_{s})dW_{s}+\int_{0}^{t}f_{2}(\cdot ,s,X_{s})ds.
\end{equation}
The Girsanov-Meyer theorem allows us to establish the existence of solution of analogous equation where the Lipschitz condition on the “drift” coefficient \(f_{2}\) is removed. Indeed if \(X\) is the solution of
(\ref{proeq5}), let \(\gamma\) be any bounded, measurable functions such that \(\gamma (\omega ,s,X_{s})\in {\bf L}\). Define
\[g(\omega ,s,x)=f_{2}(\omega ,s,x)+f_{1}(\omega ,s,x)\cdot\gamma (\omega ,s,x).\]
We will see that we can find a solution of
\[Y_{t}=Y_{0}+\int_{0}^{t}f_{1}(\cdot ,s,Y_{s})dB_{s}+\int_{0}^{t}g(\cdot ,s,Y_{s})ds\]
provided we choose a new Brownian motion \(B\) appropriately. We define a new probability measure \(Q\) by
\[\frac{d\mathbb{Q}}{d\mathbb{P}}=\exp\left (\int_{0}^{T}\gamma (s,X_{s})dW_{s}-\frac{1}{2}\int_{0}^{T}\gamma^{2}(s,X_{s})ds\right ).\]
By Theorem~\ref{prot321} we have that
\[B_{t}=W_{t}-\int_{0}^{t}\gamma (s,X_{s})ds\]
is a standard Brownian motion under \(Q\). We then have the solution \(X\) of (\ref{proeq5}) also satisfies
\begin{align*}
X_{t} & =X_{0}+\int_{0}^{t}f_{1}(\cdot ,s,X_{s})dB_{s}+\int_{0}^{t}(f_{2}+f_{1}\gamma)(\cdot ,s,X_{s})ds\\
& =X_{0}+\int_{0}^{t}f_{1}(\cdot ,s,X_{s})dB_{s}+\int_{0}^{t}g(\cdot ,s,X_{s})ds,
\end{align*}
which is a solution of a stochastic differential equation driven by a Brownian motion under \(\mathbb{Q}\).
