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Processus stochastiques

MATH 4204: Stochastic processes


The prerequisites for this course include integration and probability theory, discrete-time stochastic processes (martingales, Markov chains), and conditional expectation.


The course is divided into three parts.


First, we start with the example of the Poisson process to introduce several notions concerning continuous-time processes, including the characterization of its law with its finite-dimensional marginals, or the construction of a Markov process via a transition semigroup. We also introduce the notion of conditional distribution and review the weak and strong Markov properties.


Second, we study continuous-time Markov chains. We construct and characterize these processes (possibly up to the time of explosion), and make the connection between the transition semigroup and the matrix of intensities. We provide in particular the Kolmogorov equations and study the invariant measures and the long-time behaviour of the process.


Third, we study Brownian motion and several related processes. We start with the construction of a continuous version of the process and the study of its invariance properties and the regularity properties of its trajectories. We then dig more into the Markov and martingale properties associated with Brownian motion, and see several links with random walks, including Donsker invariance properties. Finally, we look at harmonic functions and the Dirichlet problem via Brownian motion, and if time permits continue with a glimpse of potential theory.


Responsable du MA1 :

Nikolay Tzvetkov
ENS, site Monod, bâtiment GN1 (l’arche), 4e étage

Gestionnaires de scolarité

École normale supérieure

Sophie Bonche
Site Monod, bâtiment LE (accolé au GN1, côté nord de l’allée d’Italie)
bureau 536, allée Allan C. Wilson

Université Claude Bernard

Delphine Jouve


ENS de Lyon
15 parvis René Descartes - BP 7000
69342 Lyon Cedex 07 - FRANCE
Tél. : Site René Descartes (siège) : +33 (0) 4 37 37 60 00
Site Jacques Monod : +33 (0) 4 72 72 80 00

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