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Advanced Stochastic Processes >> Content Detail



Syllabus



Syllabus



Description


The class covers the analysis and modeling of stochastic processes. Topics include measure theoretic probability, martingales, filtration, and stopping theorems, elements of large deviations theory, Brownian motion and reflected Brownian motion, stochastic integration and Ito calculus and functional limit theorems. In addition, the class will go over some applications to finance theory, insurance, queueing and inventory models.



Grading


Grading for this class will be based on the bi-weekly homework assignments, a mid-term and a final exam.



Calendar



LEC #TOPICS
1Probability Basics: Probability Space, σ-algebras, Probability Measure
2Random Variables and Measurable Functions; Strong Law of Large Numbers (SLLN)
3Large Deviations for i.i.d. Random Variables
4Large Deviations Theory (cont.) (Part 1)

Properties of the Distribution Function G (Part 2)
5Brownian Motion; Introduction
6The Reflection Principle; The Distribution of the Maximum; Brownian Motion with Drift
7Quadratic Variation Property of Brownian Motion
8Modes of Convergence and Convergence Theorems
9Conditional Expectations, Filtration and Martingales
10Martingales and Stopping Times
11Martingales and Stopping Times (cont.); Applications
12Introduction to Ito Calculus
13Ito Integral; Properties
14Ito Process; Ito Formula
15Martingale Property of Ito Integral and Girsanov Theorem
16Applications of Ito Calculus to Finance
17Equivalent Martingale Measures
18Probability on Metric Spaces
19σ-fields on Measure Spaces and Weak Convergence
20Functional Strong Law of Large Numbers and Functional Central Limit Theorem
21G/G/1 Queueing Systems and Reflected Brownian Motion (RBM)
22Fluid Model of a G/G/1 Queueing System
23Fluid Model of a G/G/1 Queueing System (cont.)
24G/G/1 in Heavy-traffic; Introduction to Queueing Networks
25Final Notes and Ongoing Research Questions and Resources

 








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