Stochastic Processes --- Fall 2002

Abstract

Markov Chains
- matrices and transition probabilities
- classification of states: irreducibility, periodicity, recurrence
- random walks
- basic limit theorems
- finding stationary distribution
Markov Processes
- Markov property
- Poisson process:
  1. memoryless property of exponential distribution.
  2. simulating a Poisson process
  3. connection with order statistics of uniform distribution.
  4. independent increments property
- Pure birth processes; general birth-and-death processes.
- Simulating a birth-and-death process.
- Finding stationary distribution of birth-and-death process (KT, p. 137).
- Probability of Extinction.
Branching Processes
- Generating function; recursive relations.
- Finding extinction probability.
Time Series
- Autocorrelation function. White noise diagnostics.
- Stationary time series. Seasonality and trend. Linear filters.
- AR and MA models. Use of ACF and PACF for model selection.
- Prediction for an AR process.
- Use of Akaike Information Criterion to choose an ARMA model.
- Non-stationary models: ARIMA, state-space models.
Brownian Motion
- Definition. Independent increments property. Markov property.
- Scaling properties. Relation to random walks.
- Reflection principle.

Table of Contents

Text

Textbooks

Introduction to Time Series and Forecasting by Peter J. Brockwell, Richard A. Davis Springer-Verlag; 2nd Bk&Cdr edition (March 8, 2002) ISBN: 0387953515
A First Course in Stochastic Processes by Samuel Karlin and Howard M. Taylor. Publisher: Academic Press; 2 edition (March 28, 1975) ISBN: 0123985528 Call Number: QA274 .K37 1975

Resourses:

conference room (Hodges Hall 4th floor), MW 1300-1430