250068 VO Stochastic processes (2020W)

A stochastic process is a sequence of random variables describing the evolution of a system changing randomly over time.

In this course we will focus particularly on Markov chains, where the future evolution of the system depends only on its current state, so that it has no "memory". Although this is a very rich class of stochastic processes (with examples arising in a huge number of applications, including physics, biology, engineering, finance and much more...) there is an elegant and powerful theory which can be used to predict its long time behaviour and fluctuations.

We will mainly develop the theory in discrete time and for discrete state spaces. We will then move on to continuous time and discuss the Kolmogorov forward and backward equations, its links with PDEs such as the heat equation (no prior knowledge required) as well as the Poisson process.
Throughout the course, the theory will be illustrated through many examples including the fundamental notion of branching processes and simple random walk. This will allow us to prove Polya's famous theorem: a drunkard moving at random will almost surely return home in two dimension, but not in three and above!

This theory requires very little prior knowledge: essentially, only a working knowledge of expectation and conditional probability on discrete spaces will be assumed, and some notions of linear algebra. In particular, no knowledge of measure theory will be assumed; in fact we view this course as an excellent way of developing a probabilistic intuition which motivates the development of a measure-based theory in further courses (such as Advanced Probability Theory).