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\begin{document}\begin{CJK*}{GBK}{}

\centerline{\heiti\bf\Large 北京师范大学~2005-2006
第一学期研究生期末考试试卷}

\bigskip

\centerline{(数学科学学院概率论与数理统计专业博士研究生~2004 级)}

\bigskip

\noindent 研究方向: 马氏过程\quad 课程名称: 随机分析\quad 任课教师:
李增沪教授

\medskip

\noindent 姓名: \qquad\qquad\quad 学号: \qquad\qquad\qquad\quad
分数:
 \hfill 日期: 2005 年~12 月~20 日

\noindent{\bf\line\line\line}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent (解题要求: 步骤清楚,  逻辑严密.)

\bigskip

\noindent \textbf{1}. Solve the following Ornstein-Uhlenbeck
equation (or Langevin equation)
 \beqnn
dX_t=\mu X_tdt + \sigma dB_t,
 \eeqnn
where the initial value $X_0$ is a constant, $\mu$ and $\sigma$ are
real constants, and $B_t$ is one dimensional Brownian motion. In
addition, find $\bfE[X_t]$ and ${\bf Var}[X_t]$.

\bigskip
\noindent \textbf{2}. Let $X$ be a bounded, measurable,
$\{\mathcal{F}_t\}$-adapted process. Let $0<T<\infty$ be fixed. We
set $X_t=0$ for $t\leq0$. Let
$\varphi_n:\mathbb{R}\rightarrow\{j2^{-n}; j=0,\pm1,\pm2,\ldots\}$
be given by
  \beqnn
\varphi_n(t)=\frac{j-1}{2^n}\;\;\;\mbox{for}\;\;\;
\frac{j-1}{2^n}<t\leq\frac{j}{2^n}.
  \eeqnn
\noindent (a) Fix $s\geq 0$. Show that $t-(1/2^n)\leq
\varphi_n(t-s)+s<t$ and that
  \beqnn
  X_t^{(n,s)}:=X_{\varphi_n(t-s)+s},\;\;\mathcal{F}_t;\quad t\geq0
  \eeqnn
is a simple, adapted process.

\noindent (b) Show that
  $
\lim_{h\downarrow0}\bfE\int_0^T |X_t-X_{t-h}|^2dt=0.
  $

\noindent (c) Use (a) and (b) to show that
  \beqnn
  \lim_{n\rightarrow\infty}\bfE\int_0^T\int_0^1|X_t^{(n,s)}-X_t|^2dsdt=0.
  \eeqnn

\noindent (d) Show that for some choice of $s\geq0$ and some
increasing sequence $\{n_k\}_{k=1}^\infty$ of integers,
  \beqnn
\lim_{k\rightarrow\infty}\bfE\int_0^T|X_t^{(n_k,s)}-X_t|^2dt=0.
  \eeqnn

\bigskip
\noindent \textbf{3}. Consider the $n$-dimensional Brownian motion
$B=(B_1\cdots, B_n)$ starting at $a=(a_1\cdots, a_n)\in\mathbb{R}$
$(n\geq 2)$ and assume $|a|<\mathbb{R}$.

\noindent (a) What is the expected value of the first leaving time
$\tau_K$ of $B$ from the ball $K = K_{\mathbb{R}} = \{x\in
\mathbb{R}^n: |x|< \mathbb{R}\}$?

\noindent (b) Suppose that $|b|>\mathbb{R}$. What is the probability
that $B$ starting at $b$ ever hits $K$\,?

\bigskip

\noindent{\bf\line\line\line}


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