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 (a more formal, yet still understandable, treatment can be found in Gardiner, Handbook of Stochastic Methods, 1985, Sect. 4.3). The fact that Brownian motion, i.e. the Wiener process, is non-differentiable, and therefore requires its own rules of calculus, is explained in Sect. 5.1. In fact, there are two dominating versions of stochastic calculus, each having advantages and disadvantages, namely the Ito stochastic calculus (Sect. 5.2), based on a pre-point discretization rule, named after Kiyoshi Ito, and the Stratonovich stochastic calculus (Sect. 5.3), based on a mid-point discretization rule, developed simultaneously by Ruslan Stratonovich and Donald Fisk. Finally, in Sect. 5.4, 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 Schrödinger equation of quantum mechanics.
KeywordsWiener Process Stochastic Calculus Stochastic Mechanic Brownian Force Forward Kolmogorov Equation
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