Stochastic analysis, spring 2010             (suomeksi)

Lecturer

Dario Gasbarra

Credits

10 op.

Type

Advanced course.

Prerequisites: Probability theory.

Language: finnish or/and english, according to the audience.

Schedule

During weeks 3-9 and 11-18, mo 10-12, tu 12-14 C124, plus 2 hours/week of exercise class. The first lecture on monday 18.1. Easter holiday on 1.-7.4.

Exams.

There will be a mid-term exam and a final exam.

Lecture notes, exercises

Content:

We will study the trheory of continuous martingales and stochastic integration.

I. Discrete time martingales. Conditional expectation, filtration and stopping times. Uniformly integrable martingales, square integrable martingales, martingale convergence theorem. Doob's maximal inequality.

II. Stochastic prosesses in continuous time. Kolmogorov's consitency theorem, construction and properties of Brownian motion. Poisson and Levy processes. Markov processes. Strong Markov property.

III. Ito calculus: functions of finite variation and Stieltjes integrals. Quadratic variation. Ito isometry for Brownian motion and Ito integral. Localization, Burkholder Davis Gundy inequality, Föllmer's pathwise inegral, Ito formula, Local time, Ito-Tanaka formula.

IV Change of measure: Girsanov theorem, stochastic exponential, Gronwall lemma. Application to stochastic filtering theory.

V Stochastic differential equations, weak and strong solutions, martingale-problem. Application: Probabilistic solutions to partial differential equations: Kakutani's theorem, Feynman-Kac formula.

VI. Ito-Clarck martingale representation theorem. Applications: option pricing in Black & Scholes financial market model.

Literature:

For the discrete-time martingale theory we follow David Williams' book: Probability with Martingales (Cambridge Mathematical Textbooks).

Karatzas, Shreve: Brownian motion and stochastic calculus, Springer 1998.

Revuz, Yor: Continuous martingales and Brownian motion, Springer 2005.

Registration

Tutorials