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 \def\blemma{\begin{lemma}\rm\fangsong{}\def\elemma{\end{lemma}}}
 \def\btheorem{\begin{theorem}\rm\fangsong{}\def\etheorem{\end{theorem}}}
 \def\bproposition{\begin{proposition}\rm\fangsong{}\def\eproposition{\end{proposition}}}
 \def\bcorollary{\begin{corollary}\rm\fangsong{}\def\ecorollary{\end{corollary}}}
 \def\bdefinition{\begin{definition}\rm\songti{}\def\edefinition{\end{definition}}}

 \def\beqlb{\begin{eqnarray}}\def\eeqlb{\end{eqnarray}}
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\begin{document}



\

\bigskip

\centerline{\huge\kaishu ��~~��~~ʦ~~��~~��~~ѧ}

\bigskip\bigskip\bigskip

\centerline{\Huge\bf ��������ҵ����(���)}

\thispagestyle{empty}

\bigskip\bigskip\bigskip\bigskip

{\Large\heiti ��ҵ����(���)��Ŀ: \qquad\qquad\qquad\qquad}

\bigskip\bigskip

\centerline{\huge\heiti\bf \underline{\qquad
Girsanov��������Ӧ��\qquad}}


\vspace{1cm}

{\large
 \begin{eqnarray*}
 \mbox{\heiti Ժ~~\,~(ϵ):} && \mbox{��ѧ��ѧѧԺ} \\
 \mbox{\heiti ר~~~~~~ҵ:} && \mbox{������������ͳ��} \\
 \mbox{\heiti ѧ~~~~~~��:} && \mbox{\qquad} \\
 \mbox{\heiti ѧ������:} && \mbox{��~~~��} \\
 \mbox{\heiti ָ����ʦ:} && \mbox{������} \\
 \mbox{\heiti ְ~~~~~~��:} && \mbox{��~~~��} \\
 \mbox{\heiti ���ڵ�λ:} && \mbox{��ѧ��ѧѧԺ}
 \end{eqnarray*}
}


\vspace{1cm}

\centerline{2002 ��\ 5 ��\ 10 ��}


\newpage

\tableofcontents


\pagenumbering{roman}


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\pagenumbering{arabic}


\newpage\addcontentsline{toc}{section}{��Ӣ��ժҪ}

\section*{��Ӣ��ժҪ}

\songti

\bigskip\bigskip\bigskip\bigskip

{\narrower{\narrower

\centerline{\large\heiti\bf Girsanov ��������Ӧ��}

\bigskip

\centerline{��~~��}

\bigskip

{\heiti ժҪ:} ����ǰ�벿�ּ������˲����˶������Ĺ�ϵ,
��������ĸ�����֪ʶ֤����\ Girsanov ����; ��벿��������\ Girsanov
�����ڷ��ٽ糬�����˶������еĳ���Ӧ��.

{\heiti �ؼ���:} �����˶�, ��, ƽ��������, �������˶�.


\bigskip\bigskip\bigskip\bigskip


\centerline{\bf\large Girsanov Theorem and its application}

\bigskip

\centerline{BAI Yun}

\bigskip

{\bf Abstract}: In the first part of this paper, the relation of
Brownian motion and martingale is discussed. Girsanov Theorem is
proved by basic techniques in probability theory. In the second
part, an application of Girsanov Theorem is given in the
construction of a critical super-Brownian motion with linear growth
rate.

{\bf Keywords:} Brownian motion, martingale, process of square
variation, super-Brownian motion.



\par}\par}




\newpage


\centerline{\huge\heiti\bf Gisanov��������Ӧ��}

\bigskip

\centerline{��~~��}


\bigskip\bigskip


\section{����}

\vspace{2mm}

\zihao{-4}

\setcounter{equation}{0}

����1827��, Ӣ��ֲ��ѧ��\ Brown
����΢���¹۲쵽�˻��������ھ�ˮ�е���ֵĲ������˶�, ����\ Wiener
������������ѧģ��, ���ڳ�֮Ϊ\ Brown �˶�. Brown
�˶������������ռ�м�����Ҫ�ĵ�λ. �����������־���\ It\^o
�Բ����˶��Ļ��ֶ����.
���������������������������������Ծ����гɹ��ķ�֧֮һ,
����ڸ��ʲ�������������滻��
������ֵı仯�������һ���������������������ʵ��Ӧ�ü�ֵ������. Girsanov
������\ It\^o �������, �õ���������\ Girsanov ����.
���Ĵӻ����ĸ�����֪ʶ������֤����\ Girsanov ����; ��������\ Girsanov
�����ڷ��ٽ糬�����˶������еĳ���Ӧ��. �йؽ������ϸ֤������\ [1] ��\
Dawson [2].



\section{��Ҫ������֤��}

��\ $(\Omega, \mcr{F}, \mbf{P})$ ��һ���걸�ĸ��ʿռ�,
$({\mcr{F}_t})_{t\geq 0}$ Ϊ�������һ��\ $\sigma$-������.

\bdefinition\label{d2.1} һά\ Brown �˶�\ $W=\{W(t), t\geq 0\}$
�Ǿ����������ʵ��������:

(1) $W(0)=0$;

(2) ��ȡ\ $0\leq t_1 \leq t_2 \leq\ldots\leq t_n$, $\{W(t_2)-W(t_1),
W(t_3)-W(t_2), \ldots W(t_n)-W(t_{n-1})\}$ Ϊ�໥���������������;

(3) ��\ $0\leq s< t$, $W(t)-W(s)$ ������̬\ $N(0,
{{\sigma}^2}(t-s))$ �ֲ�, ����\ $\sigma>0$ Ϊһ����. \edefinition

\bdefinition\label{d2.2} �����ɻ�\ $({\mcr{F}_t})_{t\ge0}$ ��Ӧ����\
$X=\{X_t, t\ge 0\}$, �������������\ $t\ge s\ge 0$, ��\
$X_s=\mbf{E}[X_t| \mcr{F}_s]$, ���\ $X$ Ϊ\ $({\mcr{F}_t})_{t\ge0}$
������. \edefinition

\blemma\label{l2.1} �����˶�\ $W$ �Լ�\ $W^2-t$ ����������. \elemma

\proof �ɲ����˶��Ķ�����, ��\ $s<t, W_t-W_s$ ��\ $\mcr{F}_s$ ����,
���
 \beqnn
\mbf{E}[W_t-W_s|\mcr{F}_s]=\mbf{E}[W_t-W_s]=0,
 \eeqnn
 \beqnn
\mbf{E}[(W_t-W_s)^2|\mcr{F}_s]=\mbf{E}[(W_t-W_s)^2]=t-s,
 \eeqnn
����
 \beqnn
\mbf{E}[W_t|\mcr{F}_s]=\mbf{E}[{(W_t-W_s)+W_s} |\mcr{F}_s]=W_s.
 \eeqnn
 \beqnn
\mbf{E}[{W_t ^2-t}|\mcr{F}_s]
 \ar=\ar
\mbf{E}[(W_t-W_s+W_s)^2-t|\mcr{F}_s] \cr
 \ar=\ar
\mbf{E}[(W_t-W_s)^2|\mcr{F}_s]+2W_s \mbf{E}[W_t-W_s|\mcr{F}_s]+W_s
^2-t \cr
 \ar=\ar
t-s+W_s ^2-t = W_s ^2-s.
 \eeqnn
�����Ķ���~2.2 ��֪\ $W$ ��\ $W^2-t$ ��Ϊ\ $({\mcr{F}_t})_{t\ge0}$ ������.
\qed

��\ $\mbf{M}_{loc}^2$ ��ʾ��ֵΪ0�ľֲ�\ $L^2$ ��ȫ��. ��\
$\mbf{M}_{loc}^c$ ��ʾ��ֵΪ0�������ֲ���ȫ��. ��\ $\mbf{L}_{loc}^2$
��ʾ����\ $\int_0^t H_s ^2<\infty$ (a.s, ���κ�\ $t\ge 0$)
֮�ɲ���Ӧ����\ $H$ ȫ��.

\bdefinition\label{d2.3} ��\ $M\in \mbf{M}_{loc}^2$, ��\
$\mbf{L}_{loc}^2(M)$ ��ʾ������������֮���Ϲ���\ $H$ ȫ��:
����ͣʱ��\ $\tau_n \uparrow \infty$ ʹ��

(1) $\tau_n$ Ϊ�ֲ���ͣʱ��;

(2) �θ�\ $ n\in N, 1_{[[\tau_n]]}H\in
\mbf{L}^2(\mbb{R}_+\times\Omega, \mcr{P}, \mu_{M^2})$. ����\
$\mu_{M^2}$ Ϊ\ $\sigma$-���޲��. \edefinition


\bdefinition\label{d2.4} �θ�\ $M\in \mbf{M}_{loc}^c$, ����\ $[M]_t
\equiv M^2 _t-2 \int_0^t{M_s dM_s}, (t\ge 0)$, ����\ $[M]=\{[M]_t,
t\ge 0\}$ ��Ϊ\ $M$ ��ƽ��������, ��\ $[M]$ ��ʹ\ $M^2-[M]$
Ϊ�����ֲ�����Ψһ����������. \edefinition

\btheorem\label{t2.1} (Brown �˶������̻�����) ��\ $X$ Ϊһά\
$({\mcr{F}_t})_{t\ge0}$ ��Ӧ����, ��\ $X_0=0$. �����������ȼ�:

(1) $X$ ��\ $({\mcr{F}_t})$-Brown �˶�;

(2) $X$ �Ǿֲ�\ $({\mcr{F}_t})$ ����\ $[X]_t=t \quad(t\ge 0)$;

(3) �Զ��κ�\ $\alpha\in R$, $Z_t^\alpha \equiv\exp\{i\alpha x_t +
\frac{1}{2} {\alpha}^2 t\}$ $(t\ge 0)$ Ϊ\ $({\mcr{F}_t})$ ��.
\etheorem


\proof $(1)\Rightarrow(2)$ ��Ϊ\ $X$ ��\ Brown �˶�, ������~2.1 ֪\ $X$
��������, ��\ $X^2-t$ Ҳ��������, ����ƽ�������̵Ķ���, $[X]_t$ ��ʹ\
$X^2-[X]_t$ Ϊ�����ֲ�����Ψһ���� ������, ��\ $[X]_t=t$.

$(2)\Rightarrow(3)$ �̶�\ $\alpha\in R$, ��\ $f(x, y)=\exp\{i\alpha
x+\frac{1}{2} {\alpha}^2 y\}$ ��\ $f_x '=i\alpha f$, $f_y' =
\frac{1}{2}{\alpha^2}f$, $f_{xx}''= -{\alpha^2} f$. ��\ It\^o ��ʽ
 \beqnn
f(X_t, t)
 \ar=\ar
f(0, 0)+ \int_0^t{{f_x '(X_s, s)} dX_s} + \int_0^t{{f_y'(X_s, s)}
ds} + \frac{1}{2}\int_0^t{f_{xx}''(X_s, s) d [X]_s} \cr
 \ar=\ar
1+i\alpha\int_0^t{Z_s ^\alpha dX_s}.
 \eeqnn
����\ $[X]_t=t$, ��\ $\mbf{E}[X]_t<\infty$, ����\ $X$ Ϊ\ $L^2$ ��.
����Ϊ\ $|Z_s ^\alpha|\leq\exp\{\frac{1}{2}{\alpha}^2 s\}$
���������� �����н�, ��\ $Z^\alpha$ Ϊ��.

$(3)\Rightarrow(1)$ ����\ $Z^\alpha$ Ϊ��, �θ�\ $ \alpha\in R,
0\leq s<t $ ��\ $\mbf{E}[Z_t ^\alpha|\mcr{F}_s]=Z_s ^\alpha$. ��
 \beqnn
\mbf{E}\Big[\exp\Big\{i\alpha X_t+\frac1 2 {\alpha}^2
t\Big\}\Big|\mcr{F}_s\Big]
 =
\mbf{E}\Big[\exp\Big\{i\alpha X_s+\frac1 2 {\alpha}^2 s\Big\}\Big].
 \eeqnn
����ͬ��\ $\exp\{i\alpha X_s+\frac{1}{2} \alpha ^2 t\}$ ��
 \beqnn
\mbf{E}[\exp\{i\alpha (X_t-X_s)\}|\mcr{F}_s] = \exp\Big\{-\frac1 2
{\alpha}^2 (t-s)\Big\},
 \eeqnn
��\ $X_t-X_s $ ����\ $\mcr{F}_s$ ��������������Ϊ��̬����\ $N(0, t-s)$
����������, ���\ $X_t-X_s$ ��\ $\mcr{F}_s$ ����, �ҷ�����̬\ $N(0, t-s)$
�ֲ�, ˵��\ $X$ Ϊ\ Brown �˶�. \qed


Ϊ��֤��\ Girsanov ����, ��������һЩ׼������:

\bdefinition\label{d2.5} �Ը����ɻ���Ӧ����\ $X=\{X_t, t\ge 0\}$,
��\ $\Re$ �϶��弯����\ $\mu_X$ ����:

(1) $\mu_X([0]\times F_0)=0$, $(F_0 \in \mcr{F}_0)$;

(2) $\mu_X([s, t]\times F)=\mbf{E}[1_F(X_t-X_s)]$, $(s<t, F\in
{\mcr{F}_0})$.

\noindent ��\ $\mu_X$ �ɿ����ڿ���\ $\sigma$ ����\ $\mcr{P}$
�ϳ�Ϊ�ɼӼ�����, ���Ϊ��\ $X$ ��\ $\mcr{P}$ �����ɵ�\ Doleans ���.
\edefinition


\blemma\label{l2.2} ����\ Brown �˶�, ��\ $\mu_{W^2}=\lambda\times
\mbf{P}$, ���е�\ $\mu_{W^2}(A\cap[[0, \tau_n]])=\mu_n(A)$ (���κ�\ $n\in
N, A\in \mcr{P}$), $\{\tau_n\}$ Ϊ����ֲ���ͣʱ��, $\mu_n$ Ϊ\
$(W^{\tau_n})^2$ ���ɵ�\ Doleans ���. \elemma


\proof �ɶ���~2.5 ��\ $\mu_{W^2}(A\cap[[0, \tau_n]])=\mu_n(A)$,
֪�����˶������ɵ��ݻ�\ $\mu_{W^2}$ �ɿ���Ϊ\ $\mcr{P}$ �ϵĲ��.

�Կ��Ͼ���\ ${(s, t]\times F}\in \Re$, $s<t, (F_0\in \mcr{F}_0). $
�ɶ���~2.5 ��
 \beqnn
\mu_{W^2}([s, t]\times F)
 \ar=\ar
\mbf{E}[1_F(W_t ^2-W_s ^2)] \cr
 \ar=\ar
\mbf{E}[1_F(W_t-W_s)^2]+2\mbf{E}[1_F(W_s(W_t-W_s))] \cr
 \ar=\ar
(t-s)\times \mbf{P}(F) \cr
 \ar=\ar
(\lambda\times \mbf{P}) ((s, t)\times F),
 \eeqnn
����\ $\lambda$ Ϊ\ $\mbb{R}_+$ ��֮\ Lebesgue ���, �ɴ˿ɼ�\ $\mu_{W^2}$
��\ $\lambda\times \mbf{P}$ ��\ $\Re$ ���غ�, �ɿ��ص�Ψһ��, ���� ��\
$\mcr{P}$ ���غ�, ������\ Brown �˶�, ��\ $\mu_{W^2}=\lambda\times
\mbf{P}$. \qed

\blemma\label{l2.3} $W$ Ϊһά\ Brown �˶�, ���\ $H=\{H_t, 0\leq t\leq
T\}$ Ϊ�ɲ���Ӧ����, ������\ $\int_0^t {H_s ^2 ds}<\infty$, ��\ $M=\int{H
dW}$ Ϊ�����ֲ���. \elemma

\proof �ɶ���~2.3 ��\ $H\in\mbf{L}_{loc}^2$, ѡȡ\ $\tau_n:= \inf
\{t:\int_0^t{{H_s}^2ds} \geq n\}\wedge n, (n\in N)$ ��Ȼ\ $\{\tau_n\}$
Ϊһ��ͣʱ, ��\ $\tau_n \uparrow \infty$. ������~2.2 ��\
$\mu_{W^2}=\lambda\times \mbf{P}$, ����
 \beqnn
\int_\Omega {\int_0^\infty{1_{[[0, \tau_N]]} H_s^2}ds}dp
 =
\int_\Omega\bigg(\int_0^{\tau_n}{H_s ^2}ds\bigg)dp\leq n,
 \eeqnn
����\ $H\in \mbf{L}_{loc}^2(W)$, ��\ $\{\tau_n\}$
Ϊ����W��H֮�ֲ���ͣʱ��, ��\ Brown �˶�WΪ�����ֲ���, ����\ ${M=\int{H
dW}}\in M_{loc} ^c$ Ϊ�����ֲ���. \qed


���²����Ǳ���Ҫ���ܵ���Ҫ����, ��\ Girsanov ����(����[1, pp. 136-137]):


\btheorem\label{t2.2} ��MΪ�����ֲ���, ��
 \beqnn
Z_t:=\exp\{M_t -\frac{1}{2} [M]_t\}, \qquad (t\ge 0)
 \eeqnn
Ϊһ�¿ɻ�����, ��
 \beqnn
\mbf{Q}:=\int {Z_{\infty}d\mbf{P}}.
 \eeqnn
��N Ϊ����\ $\mbf{P}$ �ֲ���, ��
 \beqnn
\tilde{N}_t :=N_t-[N, M]_t, \qquad (t\ge 0)
 \eeqnn
Ϊ����\ $\mbf{Q}$ �ֲ���, ��
 \beqnn
[\tilde{N}]_t ^\mbf{Q}=[N]_t, \qquad (t\ge 0)
 \eeqnn
����\ $[\tilde{N}] ^\mbf{Q} $ ��ʾ�ڲ��\ $\mbf{Q}$
�µ�ƽ��������. \etheorem

\proof (1) ��\ ${Y_t}^\alpha=\exp\{{\alpha {\tilde{N}}_t}-{{\frac1
2}{{\alpha}^2}[N]_t}\}$, ��
 \beqnn
{Y_t}^\alpha Z_t=\exp\Big\{{\alpha N_t}-{\alpha [N, M]_t}- {{\frac1
2}{{\alpha}^2}[N]_t}+M_t-{{\frac1 2}[M]_t}\Big\}.
 \eeqnn
��\ $M_t' = M_t+\alpha N_t$, $U_t' = [M_t ']$, $Y_t' = \exp\{M_t' -
{{\frac1 2}U_t'}\}$. ������~2.3 ֪\ $M$ Ϊ����\ $\mbf{P}$ �ֲ���, ��\ $M'$
Ϊ����\ $\mbf{P}$ �ֲ���. ��\ $f(x, y)=\exp\{ x-\frac{1}{2} y\}$, ��\
$f_x'= f$, $f_y'=f$, $f_{xx}'' = -\frac{1}{2} f$. ����΢�ֹ�ʽ��,
$dY_t'={{Y_t'd M_t '}-{\frac{1}{2} Y_t' d U_t'} + {\frac{1}{2} Y_t 'd
[M_t']}} = Y_t'd M_t'$. ���, $Y_t'$ Ϊ����\ $\mbf{P}$ �ֲ���, ��\
${Y_t}^\alpha Z_t$ Ϊ����\ $\mbf{P}$ �ֲ���.

(2) ��\ $(\Omega, \mcr{F}, \mbf{P})$ Ϊһ�걸�ĸ��ʿռ�, ��\
$\mbf{Q}=\int_\Omega{Z_t(H) d \mbf{P}}$, ���,
 \beqnn
Z_t={{\frac{d\mbf{Q}}{d\mbf{P}}\Big|}_ {\mcr{F}_t}}
 \qquad a.s., t\in[0, T].
 \eeqnn
�θ�\ $s<t$ ��\ $F\in \mcr{F}_s$, ���ҽ���\ $Y_t Z_t$ ����\
$\mbf{P}$ �ɻ�ʱ, $Y_t$ ����\ $\mbf{Q}$ �ɻ�, ��
 \beqnn
\int_F {Y_t d\mbf{Q}}=\int_F {Y_t Z_t d\mbf{P}},
 \eeqnn
ͬ������,
 \beqnn
\int_F {Y_s d\mbf{Q}}=\int_F {Y_s Z_s d\mbf{P}}.
 \eeqnn
��˵�\ $YZ$ Ϊ\ $\mbf{P}$ �ֲ���, $Y$ Ϊ\ $\mbf{Q}$ �ֲ���, ���� \
${Y_t}^\alpha=\exp\{{\alpha \tilde{N}_t}-{{\frac1
2}{{\alpha}^2}[N]_t}\}$ Ϊ\ $\mbf{Q}$ �ֲ���. ����ͣʱ��\ $\tau_n :=
\inf\{t:|{\tilde{N}_t|\vee[N]_t}\geq n\}$ ��ֹͣ����,

��\ $s<t$ ��\ $F\in \mcr{F}_s$ ��
 \beqnn
\int_F{\exp\Big\{{\alpha {{\tilde{N}}_s} -
\frac{1}{2}{{\alpha}^2}[N]_s}\Big\}d \mbf{Q}}
 =
\int_F{\exp\Big\{{\alpha {\tilde{N}_t} -
\frac{1}{2}{{\alpha}^2}[N]_t}\Big\}d \mbf{Q}},
 \eeqnn
�ڻ��ֺ��¶�\ $\alpha$ ����΢��, ����\ $\alpha=0$ ��,
 \beqnn
\int_F{{\tilde{N}}_s}d\mbf{Q}=\int_F{{\tilde{N}}_t}d\mbf{Q},
 \eeqnn
��
 \beqnn
\int_F({{\tilde{N}}_s ^2}-[N]_s)d\mbf{Q}
 =
\int_F({{\tilde{N}}_t ^2}-[N]_t)d\mbf{Q},
 \eeqnn
����\ $\tilde{N}$ ��\ ${\tilde{N}}^2-[N]$ Ϊ����\ $\mbf{Q}$ �ֲ���.
�ɶ���~2.4 ��\ $[\tilde{N}]$ ��ʹ \ ${\tilde{N}}^2-[N]$
Ϊ��������Ψһ����������, ��\ $[\tilde{N}]_t ^\mbf{Q}=[N]_t$. \qed


�ر�, ��\ $W$ Ϊһά\ Brown �˶�, �����¶�������.

\btheorem\label{t2.3} ��\ $W=\{W_t, 0\leq t\leq T\}$ Ϊ\ Brown �˶�, \
$H=\{H_t, 0\leq t\leq T\}$ Ϊ�ɲ���Ӧ����, ������\ $\int_0^t H_s ^2
ds<\infty$. �θ�\ $t\in[0, T]$ ��
 \beqnn
Z_t(H) := \exp \bigg\{{\int_0^T H_s d W_s}- {\frac1 2}{{\int_0^t
{H_s ^2 ds}}}\bigg\},
 \eeqnn
����\ $\mbf{E}[Z_T(H)]=1$. ��\ $\mbf{Q}=\int{Z_T (H) d\mbf{P}}$,
$\tilde{W} = W_t-\int_0^t H_s d s$ $(0\leq t\leq T)$, ��\ $\tilde{W}$ Ϊ\
$\mbf{Q}$-Brown �˶�. \etheorem

\proof (1) ��������֤\ $\{Z_t(H), 0\leq t\leq T\}$ Ϊ����������. ��\
$M=\int_0^t {H_s d W_s}$, ��\ $ U_t :=[M]_t=\int_0^t {H_s ^2 d
{[W]_s}}=\int_0^t {H_s ^2 ds}$. ��\ $f(x, y)=\exp\{x-\frac1 2 y\}$, ��\
$f_x= f$, $f_y' = -\frac{1}{2}f$, $f_{xx}'' = f$. ��\ $Z_t=f(M, U)$, ��\
It\^o ���ֹ�ʽ��΢����ʽ��
 \beqnn
d Z_t=Z_t d M_t-{\frac1 2 Z_t d U_t}+{\frac1 2 Z_t d[M]_t} = Z_t d
M_t.
 \eeqnn
����ZΪ�Ǹ������ֲ���. ��\ $\tau_n$ Ϊ��ֲ���ͣʱ��, ��\ $s<t$ ��
 \beqnn
Z_s ^ {\tau_n}=\mbf{E}[ Z_t ^{ \tau_n}|\mcr{F}_s],
 \eeqnn
��\ $n\longrightarrow \infty$, ��\ Fatou ������
 \beqnn
Z_s\geq \mbf{E}[Z_t|\mcr{F}_s], \qquad (t\in[0, T])
 \eeqnn
����ZΪ������. ��\ $Z_0(H)=1$ ��\ $\mbf{E}[Z_T(H)]=1$ ��\ $\{Z_t(H),
0\leq t\leq T\}$ Ϊһ�¿ɻ�������.

(2) �ɶ���~2.1 ����\ Brown �˶������̻���\ $[W]_t=t$, ��Ϊ\ $\tilde{W} =
W_t - \int_0^t H_s ds = W_t - [M,W]_t$, �����ɶ���~2.2 ��\ $\tilde{W}$
Ϊ����\ $\mbf{Q}$ �ֲ���, ��\ $[\tilde{W}]_t ^\mbf{Q}=[W]_t$ �� \
$[\tilde{W}]_t ^\mbf{Q}=[W]_t=t$.

(3) �������֤��\ $\tilde{W}$ Ϊ\ $\mbf{Q}$-Brown �˶� ��Ϊ\ $\tilde{W}$
Ϊ����\ $\mbf{Q}$ �ֲ���, ��\ $[\tilde{W}]_t^\mbf{Q}={[W]_t}$, �ɶ���~2.1
����\ Brown �˶������̻���, $\tilde{W}$ Ϊ\ $\mbf{Q}$-Brown �˶�. \qed



\section{Girsanov �����ļ�Ӧ��}


�������������ڳ������е�һ��Ӧ��. Ӧ��\ Girsanov
�������ǿ���ʵ���ڷ��ٽ糬�����˶� �˶�֮���ת��.

��\ $\mbb{R}^d$ Ϊ\ $d$ άʵ���ռ�.

��\ $M(\mbb{R}^d)$ Ϊ\ $\mbb{R}^d$ �����޲��ȫ��.

��\ $C([0, \infty], M(\mbb{R}^d))$ Ϊ��\ $[0, \infty)$ ��\
$M(\mbb{R}^d)$ �������������\ $w$ ȫ��.

ȡ��\ $X_t(w)=w_t$, ��\ $C([0, \infty), M(\mbb{R}^d))$ �ϵ��������.
����\ $\sigma$-������: $\mcr{G}_t=\sigma(\{X_s, s\leq t\})$.

���ǲ���֤��������������(Dawson[2]��Li-Wang[3]):

\blemma\label{l3.1} ���κ�\ $\mu \in M(R)$, ����\ $C([0, \infty],
M(\mbb{R}^d))$ ��Ψһ���ʲ��\ $\mbf{Q}_\mu$, ʹ�ö������\ $\phi\in
C^2(\mbb{R}^d)$,
 \beqnn
M_t(\phi):=\langle {X_t, \phi} \rangle-\langle {\mu, \phi}
\rangle-\frac{1}{2}{\int _0^t{\langle{X_s, \Delta\phi}\rangle ds}}
 \eeqnn
��\ $\mbf{Q}_\mu$ ��, ��ƽ��������Ϊ
 \beqnn
{{\langle M(\phi)\rangle}_t}=\int_0^t{{\langle{X_s,
{\phi}^2}\rangle}ds},
 \eeqnn
��ʱ\ $(W, \mcr{G}_t, X_t, \mbf{Q}_\mu)$ ��Ϊ�ٽ糬�����˶�.
 \elemma

�̶�\ $T>0, b\in C^2(\mbb{R}^d)$, ��
 \beqnn
\mbf{Q}_\mu ^b (d w) = \exp\bigg\{M_T (b)-\frac{1}{2}{\langle
M(b)\rangle}_T\bigg\}\mbf{Q}_\mu (dw).
 \eeqnn
����\ $(W, \mcr{G}_t, X_t, \mbf{Q}_\mu ^b)$ ��Ϊ��\ $b(\cdot)$
Ϊ���������ٶȵķ��ٽ糬�����˶�.

��������������\ Girsanov �������ǿ���֤����������

\btheorem\label{t3.1} �̶�\ $T>0, b\in C^2(\mbb{R}^d)$ ��\
$\mbf{Q}_\mu ^b$ ֮��,
 \beqnn
N_t(\phi) := \langle {X_t, \phi}\rangle-\langle {\mu, \phi}\rangle -
\frac{1}{2}{\int_0^t{\langle{X_s, \Delta\phi}\rangle ds}} -
\int_0^t{\langle{X_s, b\phi}\rangle ds}, \quad 0\leq t \leq T,
 \eeqnn
����, ��ƽ��������Ϊ
 \beqnn
{{\langle N(\phi)\rangle}_t}=\int_0^t{{\langle{X_s,
{\phi}^2}\rangle}ds}, 0\leq t\leq T.
 \eeqnn
 \etheorem

\proof ������~3.1 ��, $M_t(b), M_t(\phi), M_t(b+\phi)$ ����\ $\mbf{Q}_\mu$
��, ��
 \beqnn
\langle{{M_\cdot}(b)+{M_\cdot}(\phi)}\rangle _t=\int
_0^t{{\langle{X_s, (b+\phi)^2}\rangle} ds},
 \eeqnn
 \beqnn
[{M_\cdot}(\phi)]_t=\int_0^t{{\langle{X_s, \phi^2}\rangle} ds},
\quad [{M_\cdot}(b)]_t = \int_0^t{{\langle{X_s, b^2}\rangle} ds}.
 \eeqnn
����ƽ�������̵Ķ���
 \beqnn
[M, N]=\frac{1}{2}\{[M+N]-[M]-[N]\},
 \eeqnn
��
 \beqnn
[{{M_\cdot}(\phi), {M_\cdot}(b)}]_t
 \ar=\ar
\frac{1}{2}\Big\{[{M_\cdot}(\phi)+{M_\cdot}(b)]_t
-[{M_\cdot}(\phi)]_t - [{M_\cdot}(b)]_t\Big\} \cr
 \ar=\ar
\frac{1}{2}\Big\{\int_0^t{{\langle{X_s, (b+\phi)^2}\rangle} ds} -
\int_0^t{{\langle{X_s, \phi^2}\rangle} ds} - \int_0^t{{\langle{X_s,
b^2}\rangle} ds}\Big\} \cr
 \ar=\ar
\frac{1}{2}\Big\{\int_0^t{{\langle{X_s, (\phi ^2+2 b \phi +
b^2)}\rangle} ds} -\int_0^t{{\langle{X_s, \phi^2}\rangle} ds} -
\int_0^t{{\langle{X_s, b^2}\rangle} ds}\Big\} \cr
 \ar=\ar
\int_0^t{{\langle{X_s, b\phi}\rangle} ds}.
 \eeqnn
����Ϊ
 \beqnn
M_t(\phi) := \langle {X_t, \phi} \rangle - \langle {\mu, \phi}
\rangle - \frac{1}{2}{\int_0^t{\langle{X_s, \Delta\phi}\rangle ds}},
 \eeqnn
��
 \beqnn
Z_t := \exp\Big\{{M_t}(b)-\frac{1}{2}{\langle M(b)\rangle}_t\Big\},
\qquad b\in C^2(\mbb{R}^d),
 \eeqnn
��\ $\{Z_t, 0\leq t \leq T\}$ ��һ�¿ɻ�������, ��\ $\mbf{Q}_\mu
^b=\int_\Omega{Z_T d {\mbf{Q}_{\mu}}}$.

��\ Girsanov ����~2.2 ��,
 \beqnn
{N_t}(\phi)
 \ar=\ar
{M_t}(\phi)-[{M_t}(\phi), {M_t}(b)] \cr
 \ar=\ar
\langle {X_t, \phi} \rangle-\langle {\mu, \phi}
\rangle-\frac{1}{2}{\int_0^t{\langle{X_s, \Delta\phi}\rangle ds}} -
\int_0^t{\langle{X_s, b\phi}\rangle ds}, \quad 0\leq t \leq T,
 \eeqnn
������\ $\mbf{Q}_\mu^b$ ��, ��\ $[{{N_t}(\phi)}]_t =
[{{M_t}(\phi)}]_t = \int_0^t{{\langle{X_s, \phi^2}\rangle} ds}$.
\qed



\addcontentsline{toc}{section}{�ο�����}

\section*{�ο�����}


\begin{enumerate}


\renewcommand{\labelenumi}{[\arabic{enumi}]}

\bibitem{H01}  ��־Զ: �������ѧ����, ��ѧ������\ (2001).

\bibitem{D93} Dawson, D. A.: Measure-Valued Markov Processes. In:
{\it Lect. Notes. Math.} {\bf 1541}, 1-260, Springer-Verlag, Berlin
(1993).

\bibitem{LW99} Li, Z.H. and Wang, Z.K.: Measure-valued branching
processes and immigration processes. {\it Advances in Mathematics}
{\bf 28} (1999), 105-134.

\end{enumerate}





\end{document}
