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.


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  • Stochastic calculus for continuous processes. Martingales, local martingales, semi-martingales, quadratic variation and cross-variation, Itô's isometry, definition of the stochastic integral, Kunita-Watanabe theorem, and Itô's formula.
  • Applications to Brownian motion and martingales. Lévy characterization of Brownian motion, Dubins-Schwarz theorem, martingale representation, Girsanov theorem, and Dirichlet problems.
  • Stochastic differential equations. Strong and weak solutions, notions of existence and uniqueness, Yamada-Watanabe theorem, strong Markov property, and relation to second order partial differential equations.
  • Applications and examples.


Example Sheets

Example Classes

  • Example class 1: Mo 10 Feb, 3:30-5:30 MR3 / Tu 11 Feb, 1:30-3:30 MR3
  • Example class 2: Mo 24 Feb, 3:30-5:30 MR11 / Tu 25 Feb, 1:30-3:30 MR3
  • Example class 3: Mo 9 Mar, 3:00-5:00 MR3 / Tu 10 Mar, 1:30-3:30 MR3
  • Example class 4: Mo 20 Apr, 3:30:5:30 online / Tu 21 Apr, 1:30-3:30 online
  • Drop-in session 1: Mo 17 Feb, 4:00-5:00
  • Drop-in session 2: Mo 2 Mar, 4:00-5:00
  • Revision class: Tu 19 May, 2:00-4:00 online

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.


I will mostly follow the following references:

  • J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer, 2016.
  • D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, Springer.

I will assume the content of these references:

Lecture notes from previous years: