Category Archives: Stochastic Calculus lectures

In the study of a stochastic process it is often useful to consider some properties of the process that hold up to a random time. A natural question is for instance: How long is the process less than a given … Continue reading →

The Daniell-Kolmogorov theorem seen in Lecture 5 is a very useful tool since it provides existence results for stochastic processes. Nevertheless, this theorem does not say anything about the paths of this process. The following theorem, due to Kolmogorov, precises … Continue reading →

The Daniell-Kolmogorov extension theorem is one of the first deep theorems of the theory of stochastic processes. It provides existence results for nice probability measures on path (function) spaces. It is however non-constructive and relies on the axiom of choice. … Continue reading →

A stochastic process may also be seen as a random system evolving in time. This system carries some information. More precisely, if one observes the paths of a stochastic process up to a time , one is able to decide … Continue reading →

Let be a probability space. Definition. A (-dimensional) stochastic process on , is a sequence of -valued random variables that are measurable with respect to . A stochastic process can also be seen as an application The applications are called … Continue reading →

Stochastic processes can be seen as random variables taking their values in a function space. It is therefore important to understand the naturallly associated -algebras. Let , , be the set of functions . We denote by the -algebra generated … Continue reading →

The first stochastic process that has been extensively studied is the Brownian motion, named in honor of the botanist Robert Brown (1773-1858), who observed and described in 1828 the random movement of particles suspended in a liquid or gas. One … Continue reading →