Stochastic Calculus and Applications (Lent 2018)

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.

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:

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.