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:

1. 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.
   
2. 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.
   
3. 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.
   
4. Itô’s integral of step processes with respect to a Brownian motion, of progressively measurable processes, properties of Itô’s integral, local martingales.
   

References:

• 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.