Marie Duflo

Martingales in Discrete Time

With martingales in discrete time, we tackle all the original ideas of the theory of processes. The following chapters will therefore be either applications of this, or extensions to continuous time. They will be technically more difficult, but will be based on the same ideas.

Processes in Continuous Time

Point processes have already led us to study a random evolution at times t ∈ ℝ+. However the study has been simplified by the fact that it depends only on a sequence of r.v.’s. We are now going to deal with continuous time processes, such as Brownian motion, where the time t is an element of ℝ+. Everything that is going to be said can easily be translated by taking, instead of ℝ+, an interval of ℝ+.

Step by Step Decisions

At what time do we decide to stop playing roulette or to replace the car? When will the statistician stop his sampling? This chapter gives methods of dealing with this type of problem, where the decision taken at each instant can only be based on past experiments. These themes are regrouped under the headings optimal stopping, control or sequential statistics which, with distinct vocabularies, make use of the same ideas.

Introduction to Random Processes

As indicated by its number this short chapter is not very appealing. It contains general definitions on processes and some measure theory tools. We recommend an initial superficial reading. The definitions and terms will appear more natural after the study of some processes, in the following chapters.