ABSTRACT
This chapter is devoted to the classical theory of diffusion processes and of stochastic differential equations driven by Brownian motion. It starts with the introduction of the Brownian process as a limiting stochastic process on a time mesh sequence of independent Rademacher random variables. The Doeblin-Ito formula and its various applications in stochastic differential calculus are discussed in great details and is used to derive the Fokker-Planck equation for the density of the random states. The univariate results are extended to the case of multivariate diffusions. The scope of this chapter is to describe in full details the construction of the Brownian motion and the stochastic integrals with respect to the induced random measure on the time axis. It has chosen to present these continuous time stochastic processes from the practitioner's point of view, using simple arguments based on their discrete approximations.
