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Stochastics

These lecture notes, still a work in progress, are for a course taught at Shanghai Jiao Tong University, for graduate students.

Course Objective

This lecture aims at the evolution over time of systems that are subject to uncertainty: stochastic processes.

Throughout this lecture we will address the following fundamental concepts:

  • Probability: probability space, probability measure, expectation, measure change, conditional expectation, stochastic kernel, independence as well as fundamental inequalities and relations.
  • Discrete Martingales: discrete stochastic processes, information, stopping times, martingales and related fundamental results (convergence, law of large numbers, etc.)
  • Markov Processes: concept of memoryless stochastic processes, discrete time results, construction of the Brownian motion.
  • Ito-Integral and Calculus: construction of the stochastic integral, Ito-formula, applications.
  • Stochastic Exponential: measure change, Girsanov formula and applications
  • Stochastic Differential Equations: existence and uniqueness, strong vs weak solutions, ...
  • SDE and PDE: Relation between PDE and SDE, Kolmogorov equation, Feynman-Kak formula and applications.

Concrete Approach

The theoretical lecture combines blackboard lectures with practical applications. Lecture notes will be provided and updated during the course. The evaluation of the lecture will consists of

  • Homework: Once every two weeks to be handed out within two weeks by groups of 5-6
  • Quizz: 3-4 quizz, in class, 30 min about the past lectures.
  • Final Exam: 120 min final exam at the end of the semester.

Prerequisite

The lecture suppose that students knows about basic algebra, analysis. A previous knowledge about measure theory might be of help but nor necessary.

Literature

The lecture follows the present lecture notes. However those are inspired by uncountably many textbooks on the topic from which we present a short selection:

  • Rick Durret1: Excellent introduction to discrete stochastic processes (markov processesw and martingales and Brownian motion). No stochastic integral and SDEs
  • Steve Shreve23: Both books with a focus on Finance. The first one is discrete the second continuous and both approchable.
  • Olav Kallenberg4: Complete from probability to stochastic processes. General stochastic processes.
  • Philip Protter5: Complete about general stochastic processes (w/wo jumps). Complex.
  • Delacherie and Meyer67: The bible however machine typed and French

References

  1. Rick Durrett. Probability: Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics, 2010. ↩

  2. Steven E. Shreve. Stochastic Calculus for Finance. Volume I of Springer Finance. Springer-Verlag, New York, 2004. ISBN 0-387-40100-8. The binomial asset pricing model. ↩

  3. Steven E. Shreve. Stochastic Calculus for Finance. Volume II of Springer Finance. Springer-Verlag, New York, 2004. ISBN 0-387-40101-6. Continuous-time models. ↩

  4. Olav Kallenberg. Foundations of Modern Probability. Probability and its Applications (New York). Springer-Verlag, New York, 2nd edition, 2002. ↩

  5. Philip E. Protter. Stochastic Integration and Differential Equations. Springer, 2nd edition, 2005. ↩

  6. Claude Dellacherie and Paul-André Meyer. Probabilities and Potential. A. Volume 29 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1978. ISBN 0-7204-0701-X. ↩

  7. Claude Dellacherie and Paul André Meyer. Probabilities and Potential. B. Volume 72 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1982. ↩