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)*