Brownian motion and stochastic calculus

Brownian motion is associated with the analysis of motions whose evolution over time is so disordered that it seems difficult to predict it, even for a very short time, such as the motion of a microscopic particle suspended in a liquid and subjected to thermal agitation. More information on probability can be found in the article [AF 165], "Probabilités. Presentation", provides more details on the "invention" of Brownian motion. It plays a central role in the theory of random processes, firstly because Brownian motion is used in many applied problems to model errors or random disturbances, and secondly because Brownian motion or the diffusion processes derived from it enable the construction of simple models on which calculations can be performed.

Stochastic calculus, or Itô calculus, named after one of the pioneers in this field, is in fact a calculus of integrals with respect to Brownian motion. As Brownian motion is not a finitely-variable function, the notion of integral is not a common one, and its definition is probabilistic. In particular, it enables us to define the notion of stochastic differential equation, which is an equation obtained by the random perturbation of an ordinary differential equation. The solutions to these equations define new processes, called diffusion processes, which form the basis of modern probabilistic calculus. These processes are often Markovian, in the sense that their future behavior, conditional on the past, depends in fact only on the present state. This so-called Markov property is often verified in reality, particularly in physics, telecommunications networks or financial mathematics. Diffusion processes are therefore invaluable in modeling many random phenomena. We'll also see that there are important links between their law and certain partial differential equations. These links form the basis of many recent developments linking analytical and probabilistic results.

The first two paragraphs of this article are essential prerequisites for defining the notion of Brownian motion and understanding its properties. The probabilistic results developed in the article [AF 166] "Probabilities. Fundamental concepts". Brownian motion will then be defined using a number of approaches, enabling us to deduce its properties and demonstrate its richness. The stochastic integral is then introduced in relation to Brownian motion. The following section is devoted to the study of stochastic differential equations. Applications to the study of partial differential equations and financial mathematics are given at the end of the article.