Stochastic Differential Calculus

In this Chapter, the basic concepts of stochastic integration are explained in a way that is readily understandable also to a non-mathematician. First, we explain that Brownian motion, i.e. the Wiener process, is non-differentiable, and therefore requires its own rules of calculus. In fact, there are two dominating versions of stochastic calculus, each having advantages and disadvantages, namely the Ito stochastic calculus, based on a pre-point discretization rule, named after Kiyoshi Ito, and the Stratonovich stochastic calculus, based on a mid-point discretization rule, developed simultaneously by Ruslan Stratonovich and Donald Fisk. Finally,we illustrate the main features of Stochastic Mechanics, showing that, by applying the rules of stochastic integration, the evolution of a random variable can be described through the Schrodinger equation of quantum mechanics.