Professor: Professor R. J. Williams, AP&M 6121.
Time: M, W 4-5.20 p.m.
Place: AP&M 6438.

Office Hours: M, W 3-3.50 p.m, AP&M 6121.

DESCRIPTION: This one quarter course on stochastic processes is intended to introduce beginning mathematics graduate students and graduate students from other scientific and engineering disciplines to some fundamental stochastic processes used in stochastic modeling. For the mathematics students, this will provide valuable preparation and motivation for the more advanced graduate probability sequence, Math 280ABC. For students from other disciplines, the course will provide a theoretical basis for pursuing applied work involving stochastic models.

PREREQUISITES: Math 180A or equivalent probability course or consent of instructor.


  • Fundamental elements of stochastic processes.
  • Markov chains.
  • Hidden Markov models.
  • Martingales.
  • Brownian motion.

    TEXT: The following book will be used as a basic reference and this will be complemented with other handouts (e.g., on hidden Markov models) throughout the course.

  • G. F. Lawler, Introduction to Stochastic Processes, Chapman and Hall, New York.
    The book by Karlin and Taylor, listed below, is also a good fundamental reference, with many examples.

    General Stochastic Processes and Markov Processes:

  • S. Karlin and H. M. Taylor, A First Course in Stochastic Processes, Academic Press.
  • J. Norris, Markov Chains, Cambridge University Press, 1997.
    Hidden Markov Models:
  • P. Clote and R. Backofen, Computational Molecular Biology, An Introduction, Wiley, 2000; Chapter 5.
  • L. Rabiner and B.-H. Juang, Fundamentals of Speech Recognition, Prentice Hall, 1993; Chapter 6.
  • Iain L. MacDonald and Walter Zucchini, Hidden Markov and other Models for Discrete-valued Time Series, Chapman and Hall/CRC Press, 1997.
    Hidden Markov Models -- more advanced text:
  • R. J. Elliott, L. Aggoun, and J. B. Moore, Hidden Markov Models: Estimation and Control, Springer-Verlag, 1995.
    Brownian Motion:
  • K. L. Chung, Green, Brown and Probability, World Scientific, 1995.
    Gaussian Processes:
  • R. J. Adler, An Introduction to Continuity, Extrema and Related Topics for General Gaussian Processes, IMS Lecture Notes--Monograph Series, Vol. 12, 1990.
  • S. M. Berman, Sojourns and Extremes of Stochastic Processes, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992.
    Markov Chain Monte Carlo Simulation Methods:
  • C. Robert and G. Casella, Monte Carlo Statistical Methods, Springer, 1999.
  • C. Robert, Discretization and MCMC Convergence, Lecture Notes 135, Springer, 1998.
    Markov Decision Processes
  • M. L. Puterman, Markov Decision Processes, Wiley, 1994.
  • E. Altman, Constrained Markov Decision Processes, Chapman and Hall, CRC Press, 1999.
    Fitting Stochastic Models to Data:
  • E. A. Thompson, Statistical Inference from Genetic Data on Pedigrees, NSF-CBMS Regional Conference Series in Probability and Statistics, Vol. 6, Institute of Mathematical Statistics, 2000.
  • B. J. T. Morgan, Applied Stochastic Modelling, Arnold Publishing, London, 2000.
    Selected Applications of Stochastic Modeling:
  • P. Baldi and S. Brunak, Bioinformatics: The Machine Learning Approach, MIT Press, Second Edition, 2001.
  • R. Durbin, S. Eddy, A. Krogh, and G. Mitchison, Biological Sequence Analysis, Cambridge University Press, 1998.
  • M. S. Waterman, Introduction to Computational Biology, Chapman and Hall/CRC Press, 1995.
  • G. Winkler, Image Analysis, Random Fields, and Dynamic Monte Carlo Methods, Springer, 1995.


  • Hidden Markov Model Matlab Toolbox (GNU software)


  • Web sources on hidden Markov models (under construction)
  • Web sources on bioinformatics (under construction)
  • Brownian motion: a description of aspects of this process and some of its applications (provided by Y.K.Lee and Kelvin Hoon)
    Contact information: If you are interested in this course, please send email to stating your background in probability, your department and any questions you might have.