** Random process **

** ENPC, 2nd year.**

**In charge:**
*Jean-François
Delmas*
** et **
*Axel Parmentier*

**Polycopié**
by *A. Alfonsi*.

**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
(Chapter I).
- Course 2: II Conditional expectation
(Chapter II)
(exercises).
- Course 3: III Markov chains (i)
(exercises).
- Course 4: III Markov chains (ii)
(exercises).
- Course 5: III Exercises (Chapter
III).
- Course 6: IV Martingales (i).
- Course 7: IV Martingales (ii).
- Course 8: IV Exercises
(exercises).
- Course 9: V Optimal stopping time
(Chapter V).
- Course 10: VI Brownian motion (i).
- Course 11: VI Brownian motion (ii).
- Course 12: VII An application: TBA.
- Course 13: Exam. Previous exams:
2016,
2017,
2018.

**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.