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: S. Péché
Tutorials: B. Laslier
Périod: Term 1
Hourly volume: 3 hours of lectures and 3 hours of tutorials per week

Stochastic processes are well suited for modeling the evolution of a dynamical system when the evolution cannot be predicted with certainty from the initial state of the system and from an evolution equation. The surrounding world offers a wide variety of such phenomena: population dynamics, concentration of particles recorded by a counter, stock prices, ...

The course begins with the definition and study of Brownian Motion which occupies a central place in the theory of random processes. We focus on the study of a certain number of its properties. It will continue with an introduction to stochastic calculus (Ito’s integration) which has proven for several years to be a remarkable tool in the study of stochastic processes. 
Finally, the course will end with the study of  diffusions described by stochastic differential equations, which model a good number of physical phenomena: deterministic evolution perturbed by some random noise, evaluation of conditional assets, etc …


-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)