Martingale Theory (MSF400/MVE315)

läsperiod 2, 2010

OBS! Updated 27/10 Schedule week 1 (27/10 and 28/10):

OBS! Updated 27/10 Schedule week 2 and after:

  • monday 15:15-17:00 MVL:15
  • wednesday 15:15-17:00 MVL:15

OBS! Updated 6/12!!! Schedule week 7 (6/12 and 7/12):

  • monday 15:15-17:00 MVL:15
  • tuseday 15:15-17:00 MVL:14 Observe the changed day and place!!!

Teacher and examiner Erik Broman

Exam
Written. 17th of december, 8:30-13:30 in the V-house.

Purpose of course
To gain basic knowledge of discrete Martingales. The theory of Martingales is an extremely useful tool used in Stochastic Processes, Financial Mathematics, Branching Theory, Diffusion Theory and Partial differential to name a few! If time permits we might also spend some time on continuous martingales.

Prerequisites

  • A basic course in probability covering concepts such as probability distributions, expectations and variance.
  • Some course in measure and integration theory. For example ``Probabilities and Expectations'' or ``Measure and Integration theory''.

    • In addition, general knowledge about probability theory is useful but not required. For instance courses such as

  • Markov theory
  • Foundations of probability theory

    • are (might be) useful.

    What will we do?

    Material that will definately be covered:

  • Conditional Expectations
  • Martingales
  • Doobs Optional Stopping Theorem
  • Convergence of Martingales
  • L^2-bounded Martingales
  • Kolmogorov's Strong Law of Large Numbers
  • Uniform Integrability (Uniformly Integrable Martingales)

    • In addition, we will cover some or all of:

  • Option pricing with the discrete Black-Scholes formula
  • The Mabinogian sheep problem
  • Applications to branching processes
  • Continuous time martingales

    • How many and which of the above bullets that will be presented depends on:

  • The amount of time disposible
  • Interest from course participants

    • Who should take this course?
    Everyone. In particular:

  • Any PhD student in probability theory/statistics.
  • Any advanced masters student that has an interest in probability theory and/or statistics.
  • Any PhD student in mathematics who wants to know more about basic, important concepts of probability theory.

    • Organisation
    The course will consist of lectures. In addition there might be home assignements and/or participant lectures.

    Litterature
    D. Williams: Probability with martingales, Cambridge Mathematical Textbooks Cambridge University Press, Cambridge, 1991. ISBN: 0-521-40455-X; 0-521-40605-6