The major interest of Dr. Chiang's are Probability theory, Stochastic processes and especially, their applications to Statistics and Mathematical Analysis.

 For the system of $d-$dimesnional stochastic differential equations, $d\ge 2$,
$$ \begin{array}{ll} & dX_t^{\epsilon} =b(X_t^{\epsilon})dt+\epsilon dW_t,\ \ \ \ t\in [0,T]\\ & X_0^\epsilon = x \in H\subseteq R^d, \end{array} $$

 where $b(x)=(b_1(x),...,b_d(x))$ is a bounded smooth vector field except along the hyperplane $H=\{ x\in R^d, x_1=0\}$ but satisfies the stability condition in the sense that there exists a constants $c>0$ such that for some $\delta_0>0$ $b_1(x)\le -c$ if $x_1\in (0,\delta_0)$ and $b_1(x)\ge c$ if $x_1\in (-\delta_0,0)$, we shall prove that the central limit theorem holds for $(X^\epsilon (\cdot), u^{\epsilon+}(\cdot))$ where $u^{\epsilon+}_t$ is the  occupation time of $X^\epsilon(t)$ in the right half space $H^+=\{ x\in R^d, x_1>0\}$ up to time $t$. To be explicit, we shall show that there exist two continuous functions $\phi (\cdot) \in C([0,T],R^d)$ and $\psi(\cdot)\in C([0,T],R)$ such that  the process $(\frac{1}{\epsilon}(X^\epsilon(\cdot)-\phi(\cdot)), \frac{1}{\epsilon}(u^{\epsilon+}{(\cdot)}-\psi(\cdot)))$ converges  to a Gaussian proess in $C([0,T],R^{d+1})$ in probability  thus in distribution as $\epsilon\to 0$.

Research Fields: Probability

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