Math 632 - Introduction to Stochastic Processes

• Prerequisites: Math 431, or Stat 309 and 310, or Stat 311 and 312, or Stat 313 or 314.
• Frequency: Fall (I), Spring (II)
• Student Body: advanced undergraduates and graduate students
• Credits: 3. (N-A)
• Recent Texts: Introduction to Stochastic Processes by Hoel, Port, and Stone, or Essentials of Stochastic Processes by Durrett
• Course Coordinator: Jim Kuelbs
• Background and Goals: Math 632 gives an introduction to Markov chains and Markov processes with discrete state spaces and their applications. Particular models studied include birth-death chains, queuing models, random walks and branching processes. Selected topics from renewal theory and Brownian motion are also included, but vary from semester to semester to meet the needs of different audiences.
• Alternatives: n/a
• Subsequent Courses: Math 635, 735, 831-832

Content coverage:

• Markov Chains
  • transition functions and related computations
  • classification of states: recurrence, transcience, irreducibility, periodicity
  • examples: queuing, birth-death chains, branching, random walks
• Limiting Behavior of Markov Chains
  • the main limit theorem and stationary distributions
  • absorption probabilities
  • further recurrence criteria
• Continuous Time Markov Chains
  • definitions and examples (Poisson process)
  • structure of a Markov process: waiting times and jumps
  • the Kolmogorov differential equations
  • limit theory
  • birth-death processes and other examples
• Selected Topics
  • renewal theory and applications
  • a first look at Brownian motion and some applications