Mathematical background on stochastic processes

Abstract

As mentioned in the introduction to this volume, the purpose of the formation was mainly to provide an in-depth presentation of stochastic processes as modelling tools. The chosen standpoint is thus a physical point of view. However, it has also been emphasised that, in order to be able to follow the details of specific models and build bridges between different subjects where new ideas related to stochastic modelling can appear, researchers must have a sound knowledge of the mathematical properties of stochastic processes. This represents a middle-of-the-road approach. Indeed, even though the subject is still relatively young, a vast mathematical literature exists on stochastic processes (Arnold, 1974; Klebaner, 1998; Oksendal, 1995; Karatzas and Shreve, 1991) but these works may not be easily accessible to physically-oriented readers. On the other hand, stochastic processes have been used in separated fields of Applied Physics but not always with a clear presentation or resorting to some ‘recipes’. Yet, in recent decades, attempts have been made to come up with improved introductions to stochastic processes in the Physics community.