Retired Research Fellow |   Chiang, Tzuu-Shuh
Contact Information
• matsch$\color{red}{@}$math.sinica.edu.tw
• +886 2 2368-5999 ext. 610
• +886 2 2368-9771
Research Interests
• Probability
Education
• Ph.D. The University of Minnisota (1980)
• M.S. University of Wisconsin-Milwaukee (1975)
• B.S. National Taiwan University (1972)

Work Experience
• visiting member University of North Carolina 1989 - 1991
• Research Fellow Institute of Mathematics, Academia Sinica, R.O.C. 1986 - Present
• Visiting associate professor University of Minnisota 1985 -1986
• Associate Research Fellow Institute of Mathematics, Academia Sinica, R.O.C. 1981 - 1986
• Assistant Professor University of Wisconsin-Milwaukee 1980 - 1981

Research Descriptions

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 $\epsilon\to 0$.

Selected Publications
1. "(with S. J., Sheu) The Central Limit Theorem for Diffusion Processes with a Small Parameter in Discontinuous Media" , Frontiers of Chinese Mathematics , 2015-06.
2. (with Sheu, Shuenn-Jyi) "A Convexity Approach to Option Pricing with Transaction Costs in Discrete Models" , Stochastic Analysis with Finacial Application , 65, 300-319, 2011-06.
3. (with Chow,Yunshyong) "Optimal Ventcel Graphs, Minimal Cost Spanning Trees and Asymptotic Probabilities," , Applicable Analysis and Discrete Mathematics , 1, 1-10, 2007-04.
4. (with Shiu,Shang-Yuan, Sheu, Shuenn-Jhi) "(with S.J.Sheu and S.Y.Shieu) Price systems for markets with transaction costs and control problems for some finance problems" , IMS Lecture Notes-Monograph series Time series and Related Topics , 52, 257-271, 2006-07.
5. (with S. J., Sheu) "Small perturbation of diffusions in inhomogeneous media" , Annales de L'I.H.P., Prbas and Stats. , 38 (3), 285-318, 2002.
6. (with S. J., Sheu) "Large deviation of diffusion processes with discontinuous drift, Stochastic Analysis and Related Topics VII, The Silivri Workshop. Birkh$\ddot{a}$user" , 159-175, 2001.
7. (with Y. Chow and J. Hsieh) "Comparison of Metropolis algorithm and Swendsen-Wang dynamics" , Proceedings of 3rd Sino-Philippine Symposium in Analysis Matimras Matematika, special issue , 21-31, 2000.
8. (with Y. Chow) "On the exponential convergence of some inhomogeneous Markov chains" , Soochow J. of Mathematics , 26 (2), 97-101, 2000.
9. (with Shuenn-Jyi Sheu) "Large deviation of diffusion processes with discontinuous drift and their occupation times" , Ann. of Probab. , 28, 140-165, 2000.
10. (with Yunshyong Chow) "Asymptotic behavior of some linear differential systems" , Journal of Diff. Equ. , 144 (2), 321-352, 1998.
11. (with Yunshyong Chow) "On the exit problem from a cycle of simulated annealing processes -- a backword equation approach" , Annals of Appl. Prob. , 8 (3), 896-916, 1998.
12. (with Shuenn-Jyi Sheu) "Large deviation of small perturbation of some unstalbe systems" , Journal of Stochastcic Analysis and Applications , 15 (1), 31-50, 1997.
13. (with Yunshyong Chow) "Simulated annealing and optimal cooling rates for some 1-dim spin glass models" , Proceedings of Symposia in Pure Mathematics AMS , 83-93, 1995.
14. (with Yunshyong Chow) "On global and sharp Markov properties of random functional series in Sobolev spaces" , Theory of Probability and its applications (Russian edition) , 40 (2), 464-471, 1995.
15. (with Yunshyong Chow and Y. J. Lee) "Exact formulas of certain functional integrals on Wiener space" , Stochastics and Stochastic Reports , 50, 211-223, 1994.
16. (with Yunshyong Chow) "The asymptotic behavior of simulated annealing processes with absorption" , SIAM J. Control and Optimization , 32 (5), 1247-1265, 1994.
17. "Mckean-Vlasov equations with discontinuous coefficients" , Soochow Journal of Mathematics , 20 (4), 507-526, 1994.
18. (with Yunshyong Chow) "Asymptotic behavior of eigenvalues and radom updating schemes" , Applied Math. and Optimization , 28, 259-275, 1993.
19. (with Yunshyong Chow, June Hsieh) "Some limit theorems on simulated annealing" , Lecture Notes in Statistics, Springer-Verlag , 125-131, 1992.
20. (with Yunshyong Chow) "A comparison of simulated annealing of Gibbs sampler and Metropolis algorithms" , Lecture Notes in Statistics, Springer-Verlag , 74, 117-124, 1992.
21. (with G. Kallianpur, P. Sundar) "Propagation of chaos for systems of interacting neurons" , Pitman Research Notes in Mathematics Seriees , 268, 98-110, 1992.
22. (with G. Kallianpur and P. Sundar) "Propagation of chaos and the Mckean-Vlasov equation in duals of nuclear spaces" , Applied Math. and Optimization , 24, 55-83, 1991.
23. (with Yunshyong Chow) "A limit theorem for a class of inhomogeneous Markov processes" , Ann. Probability , 17, 1483-1502, 1989.
24. (with Yunshyong Chow) "On the convergence rate of annealing processes" , SIAM J. Control Optimization , 26, 1455-1470, 1988.
25. (with Shuenn-Jyi Sheu) "(With ) Large deviations of infinite dimensional Ornstein- Uhlenbeck process on C[0, 1]" , Stochastics , 23, 159-178, 1988.
26. (with Yunshyong Chow) "On eigenvalues and annealing rates" , Mathematics of Oper. Research , 13, 508-511, 1988.
27. (with Chii-Ruey Hwang and Shuenn-Jyi Sheu) "Diffusion for global optimization in $R^n$" , SIAM J. Control Optimization , 25, 737-757, 1987.
28. (with Yunshyong Chow and Y.J. Lee) "A formula for $E_W\,{\rm exp}(-2^{-1}a^2|x+y|^2_2)$" , Proc. Amer. Math. Soc. , 100, 721-724, 1987.
29. (with Yunshyong Chow) "On the convergence rate of a special class of annealing processes" , Soochow J. Math. , 13, 23-30, 1987.
30. "On the lower bound of large deviation of random walks" , Ann. Probability , 1, 90-96, 1985.
31. (with Chii-Ruey Hwang) "On the non-uniqueness of the limit points of diffusions with a small parameter" , Stochastics , 10, 149-153, 1983.
32. "Large deviations of some Markov processes on compact metric space" , Z. Wahrsch. Verw. Gebiete , 61, 71-81, 1982.
33. "A lower bound of the asymptotic behavior of some Markov processes" , Ann. Probability , 10, 955-967, 1982.
34. "A note on the irreducibility of Lebesque measure with applications to random walks on the unit circle" , Proc. Amer. Math. Soc. , 85, 603-605, 1982.