Chapter 4 - Stochastic processes

Abstract

This chapter presents a discussion on stochastic processes. The notion of stochastic process is used to describe the random time dependent variables. The Wiener process (or the Brownian motion) plays the central role in mathematical finance. The chapter briefly explains the Markov process and Chapmen-Kolmogorov equation. The Markov process represents the next level of complexity, which embraces an extremely wide class of phenomena. In this process, the future depends on the present but not on the past. This means that the evolution of the system is determined with the initial condition. The notions of the generic Wiener process and the Brownian motion are sometimes used interchangeably, though there are some fine differences in their definitions. The Brownian motion is the classical topic of statistical physics. In mathematical statistics, the notion of the Brownian motion is used for describing the generic stochastic process. Yet, this term referred originally to Brown's observation of random motion of pollen in water. Random particle motion in fluid can be described using different theoretical approaches. The Einstein's original theory of the Brownian motion implicitly employs both the Chapman-Kolmogorov equation and the Fokker-Planck equation. However, choosing either one of these theories as the starting point can lead to the diffusion equation.