| Lectures: | S. Péché |
| Tutorials: | B. Laslier |
| Period: | Term 1 |
| ECTS: | 6 |
| 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 …
Program:
-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)
-Diffusions.
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)
