Course title: Stochastic calculus 1 (Calcul stochastique 1) Teacher: B. De Meyer

UE 1

Major(s): FQ

36 hours (5 ECTS)

Evaluation: Written exam and possibly a complementary oral exam.

Prerequisites:

Advanced probability theory Presentation:

Stochastic Calculus is the mathematical toolbox of finance. The course of Stochastic calculus 1 is a mathematically founded presentation of the main concepts needed to introduce Itô’s integral with respect to a Brownian Motion. It is taught in 4 weeks, 9 hours per week. This course is the prerequisite of Stochastic calculus 2.

Details of the sessions:

UE 1

Major(s): FQ

36 hours (5 ECTS)

Evaluation: Written exam and possibly a complementary oral exam.

Prerequisites:

Advanced probability theory Presentation:

Stochastic Calculus is the mathematical toolbox of finance. The course of Stochastic calculus 1 is a mathematically founded presentation of the main concepts needed to introduce Itô’s integral with respect to a Brownian Motion. It is taught in 4 weeks, 9 hours per week. This course is the prerequisite of Stochastic calculus 2.

Details of the sessions:

- Reminder on probability theory: sigma-algebra, monotone class theorem, probability, con- ditional probability, independence, expectation of random variables, characteristic function, convergence of random variables, gaussian vectors, conditional expectation as an orthogonal projection, properties of the conditional expectation.
- Stochastic processes, basic definitions, Brownian motion definition, construction the Brow- nian motion, Kolmogorov’s theorem, the Wiener space, properties of the Brownian motion, quadratic variation.
- Stopping times, progressively measurable processes, discrete time martingales, Optional stop- ping theorem, Doob’s inequatity, Continuous time martingales, the space of square integrable continuous martingales and its completeness, Uniform integrability.
- Itô’s integral of step processes with respect to a Brownian motion, of progressively measurable processes, properties of Itô’s integral, local martingales.

- Revuz, D., M. Yor, Continuous Martingales and Brownian Motion, Springer 2005.
- Karatzas, I., S. Shreve, Brownian Motion and Stochastic Calculus, 2nd. ed., Springer 1991.
- Lamberton, D., B. Lapeyre, Introduction to Stochastic Calculus Applied to Finance, 2nd. Ed., Chapman and Hall, 2007.