Topic I: Introduction to Stochastic Calculus and Applications

 

Lecturers: 姜祖恕 (中研院數學所退休研究員)、許順吉 (中央大學數學系)

TA: 陳韋達 (中央大學數學系博士生)

 

Contents:

 

Brownian Motion

Brownian motion as Gaussian process

Path properties

Markov Properties

Martingale properties

Others Strong Markov properties

Skorokhod imbedding

Donsker theorem

Stochastic Integration

Quadratic variation

Ito’s integral

Ito’s formula

Stochastic differential equations

Applications:

Dirichlet Problem

Hormander condition and smoothness of transition density

Option Pricing

Investment Problem

 

Prerequisite:

 

Basic mathematical analysis and some experience with measure theory approach of probability theory will be helpful. The book by David Williams will help you to develop some ideas about martingales and some of their applications.

 

References:

 

Topic II: Markov chains and mixing times

 

Lecturer: 陳冠宇(交通大學應數系)

 

Abstract:

 

This course is intended to build up a bridge from the classical theory of Markov chains to the quantitative analysis of mixing times. After a quick review on the fundamental result, we will turn to the analysis of convergent behavior. Concerning the mixing times, we will focus on the measurements of total variation and separation, and explore the coupling time and strong stationary time. For an illustration, we will introduce several classical examples, including the ruin problem, the coupon collection and the card shuffling.

 

Prerequisite:

 

Undergraduate probability theory.

 

Course perspective:

 

After the lectures, we encourage students to go through the notes, which will be distributed during speeches, and to join the informal discussion on related topics and open problems.