M2MO: Modélisation Aléatoire, Finance et Data Science

Master en statistique, probabilités et finance - Université Paris 7 - Paris Diderot

Courses Core courses Stochastic calculus and diffusion models

Stochastic calculus and diffusion models

Lectures: C. Labbé 
Tutorials: S. Coste 
Period: Term 1
Hourly volume: 3 hours of lectures and 3 hours of tutorials per week


Stochastic processes in continuous time arise in many different situations (stock prices, population dynamics, evolution of particles at large scale). This course aims at introducing the theory of stochastic calculus which allows to provide very detailed information on the behavior of such processes.




The course will start with the definition and study of the Brownian motion, the most important process in continuous time. Then we will introduce some elements of the theory of continuous martingales, and this will allow us to define Itô’s integral. The end of the course will be concerned with stochastic differential equations and diffusion processes.


-Gaussian random variables, Gaussian spaces
-Brownian Motion (Makov’s property)
-Itô calculus
-Martingales and Girsanov theorem.
-Stochastic Differential Equations (basic equations, existence and uniqueness of the solutions)

Bibliography :

  • CHUNG ET WILLIAMS Introduction to stochastic integration, Birkhaüser (1983).
  • I. KARATZAS, S. SHREEVE Brownian motion and stochastic calculus, Springer (1998)
  • B. OKSENDAL Stochastic differential Equations, Springer, Fifth Edition (1998)
  • D. REVUZ, M. YOR Continuous martingales and brownian motion, Springer, Third Edition (1999)