Stochastic processes and applications
ENPC, 2nd year.
In charge:
Jean-François
Delmas,
Guillaume Chennetier
and
Sophian
Mehalla.
Notes (pdf) and
Exercises (pdf).
Goals
Our goal is to present some very common random process in a discrete
setting such as Markov chains and in a continuous setting such as
Brownian motion. We present a detailled introduction to Markov chains
(discrete time and discrete space) which are natural models for
stochastic dynamical systems, to martingales (discrete time) which are
powerful tools to study random processes, with an application to the
optimal stopping (discrete time): when to stop playing in a game in
order to maximise the expected gain. We end with an introduction to
Brownian motion.
Program:
- Course 1: I Measure theory and integration.
- Course 2: II Conditional expectation
- Course 3: III Markov chains (i)
- Course 4: III Markov chains (ii)
- Course 5: III Exercises.
- Course 6: IV Martingales (i).
- Course 7: IV Martingales (ii).
- Course 8: IV Exercises
- Course 9: V Optimal stopping time.
- Course 10: VI Brownian motion (i).
- Course 11: VI Brownian motion (ii) and exercises.
- Course 12: Exam. Previous exams:
2016: Q-process.
2017: Brownian bridge. Galton-Watson
process.
2018: Distribution of the maximum of
a random walk.
2019: House of cards. Yet another
representation of the Brownian bridge.
2020: Mean of exit time. When is a functional of a
Markov chain again a Markov chain?
2021: Characterization of
martingales. Polya urns. Markov Gaussian process.
2022: Random walk in random environment.
2023: de Finetti theorem. Polya urns
(again!).
2024: Lindley process. Lyapounov functions.
2025: Renewal process. Optimal
parking place.
Main References
- P. Baldi, L. Mazliak et P. Priouret. Martingales et chaînes de
Markov. Hermann (éditions), 2001.
- M. Benaïm et N. El Karoui. Promenade aléatoire. École
Polytechnique, 2004.
- J.-F. Delmas et B. Jourdain. Modèles aléatoires. Math. et
Appli., SMAI, Springer, 2006.
- R. Durrett. Probability: Theory and Examples (4th edition).
Cambridge Series in Statistical and
Probabilistic Mathematics, 2010.
- A. Klenke. Probability Theory. Springer, 2014.
- D. Williams. Probability with Martingales.
Cambridge Mathematical Textbooks, 1991.
Past teacher: