Stochastic Calculus and Applications (Lent 2020)

Time and Location: M-W-F, 9-10am; lectures in room MR5; example classes see below.

Lectures: Roland Bauerschmidt rb812, Example classes: Daniel Heydecker dh489

This course is an introduction to Itô calculus, in Part III of the Cambridge Tripos.

COVID-19 Update on the Exam

The details of the online open book pass/fail exam in June are still being worked out and further information will be provided or linked here as it becomes available. I expect that the level of the questions will be comparable to that of the more straightforward problems in previous years.

Schedule

Notes

Example Sheets

Example Classes

Approximate list of lectures

  1. Introduction. The Wiener Integral.
  2. Lebesgue-Stieltjes integration and signed measures.
  3. Lebesgue-Stieltjes integration and signed measures.
  4. Finite variation processes.
  5. Local martingales.
  6. L^2 bounded martingales.
  7. Quadratic variation.
  8. Quadratic variation, covariation.
  9. Kunita-Watanabe inequality, semimartingales, Ito integral for simple processses
  10. Ito isometry
  11. Consequences of Ito isometry
  12. Integration by parts and Ito formula
  13. Levy's characterisation of Brownian motion. Dubins-Schwarz theorem.
  14. Dubins-Schwarz theorem. Girsanov's theorem.
  15. Girsanov's theorem and Cameron-Martin formula.
  16. SDEs
  17. Strong existence of solutions.
  18. Solution map.
  19. Examples of SDEs and local solutions.
  20. Local solutions and Dirichlet-Poisson problem.
  21. Dynkin's formula. Cauchy problem.
  22. Markov property.
  23. Generator and invariant measures.
  24. Convergence to equilibrium.

References

I will mostly follow the following references:

I will assume the content of these references:

Lecture notes from previous years: