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06 - 15
[ Wed. ]
2016 NCTS Probability Summer Courses
Topic I: Introduction to Stochastic Calculus and Applications
Topic II: Markov Chains and Mixing Times
 
Registration:

 

 

Date:

Topic I: July 25- Aug. 5 (Mon- Fri)
Time: 10:00-12:00 Lectures; 13:00- TA session
 
Topic II: Aug. 9, 11, 16(Tu, Th, Tu)
Time: 10:00-12:00 Lectures; 13:00-14:00 Informal discussion.
 
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:
  1. Thomas M. Liggett, Continuous Time Markov Processes: An Introduction.
  2. David Williams, Probability with Martingales
  3. D. Lamberton and B. Lapeyre, Introduction to Stochastic Calculus Applied to Finance
 
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.
 
 

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