奇異擾動擴散的漸進常態

Asymptotic Normality for Singularly Perturbed Diffusion Processes

陳韋達(Wei-Da Chen)

指導教授 ： 姜祖恕

共同指導教授 ： 張志中

國立臺灣大學/理學院/數學研究所/碩士(2010年)

https://doi.org/10.6342/NTU.2010.03011

摘要

給定一個隨機擴散過程。這個擴散過程具有兩個時間尺度。一個是改變極快速的尺度，而另一個是比較慢的尺度。在本論文裡，我們有興趣的是擴散過程的停留時間之函數當ε →0。在我們的直覺中，當ε →0我們認為這個擴散過程會被比較快的部份所控制。為了使我們的直覺更加明確，我們使用這個擴散過程的機率密度函數的逼近式去估計當ε →0時的行為。用以這個擴散過程的機率密度函數的逼近式，我們將證明這個擴散過程的停留時間之函數的大數法則以及漸進常
態。

關鍵字

奇異擾動擴散

並列摘要

Let Xε (·) be a diffusion process satisfying. This diffusion process has two time scales. One is a rapidly changing scale, and the other is a slowly varying scale. In this paper, we are interested in a function of the occupation time of when ε → 0. In our intuition, we think this diffusion will be driven by its fast part when ε → 0. To make our intuition more precisely, we use the asymptoticity for the density of this diffusion to estimate its behavior when ε →0. By virtue of asymptoticity for the density of this diffusion, we will show the law of large
numbers and the asymptotic normality of a function of the occupation time of this process.

並列關鍵字

Singularly perturbed diffusion processes

參考文獻

[2] A.BENSOUSSAN, Perturbation Methods in Optimal Control, Wiley, Chichester, 1988.

[3] G. YIN, Q. Zhang, Continuous Time Markov Chains and Applications.

[4] H. J. Kushner, Approximation and Weak Convergence Methods for Random Processes,

[5] D.G. ARONSON, Non-negative solutions of linear parabolic equations, Ann. Scuola

[1] R.Z.KHASMINSKII and G. YIN, Asymptotic series for singularly perturbed