Revision classes
The revision classes will take place on Mo 28 May and Tu 29 May, 2-4pm in MR14.
Daniel will go through the question from 2016-2017.
Time and Location: M-W-F, 9-10am; lectures in room MR5; example classes in room MR14
Lectures: Roland Bauerschmidt rb812, Example classes: Daniel Heydecker dh489
This course is an introduction to Itô calculus, in Part III of the Cambridge Tripos.
Syllabus
- 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, conformal invariance of planar Brownian motion, 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.
Stroock-Varadhan theory. Diffusions, martingale problems, equivalence with SDEs, approximations of diffusions by Markov chains.
Example Sheets
The example sheets will be posted here during the course of the term.
- Example Sheet 1 (posted January 19). Questions to hand in: 7 and 12 (without bonus). Solution sketch.
- Example Sheet 2 (posted February 2). Questions to hand in: 3 and 5. Solution sketch.
- Example Sheet 3 (posted February 16). Questions to hand in: 3 and 6. Solution sketch.
- Example Sheet 4 (posted March 3). Questions to hand in: 4 and 7. Solution sketch.
The example classes will take place in MR14. The tentative dates are:
- Th 8 Feb, Fr 9 Feb, 1-3pm
- Th 22 Feb, Fr 23 Feb, 1-3pm
- Th 8 Mar, Fr 9 Mar, 1-3pm
- Mo 28 May, Tu 29 May, 2-4pm
References
I will mostly follow the following references:
- J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Springer, 2016.
- N. Berestycki, Stochastic Calculus.
- M. Tehranchi, Stochastic Calculus.
- V. Silvestri, Stochastic Calculus.
- J. Miller, Stochastic Calculus.
- J. Norris, Advanced Probability.
- P. Sousi, Advanced Probability.
- J. Norris, Probability and Measure.