Introduction: Origin of Stochastic Differential Equations

Abstract

Differential equations play a central role in applications of mathematics to natural and engineering sciences. However, differential equations governing real processes always contain some elements (e.g. coefficients, inhomogeneous part) which characterize physical features of the phenomenon and environment and are experimentally determined. Due to errors in the measurements and inherent randomness of a phenomenon these elements cannot most often be expressed by one uniquely defined function f(x), but they have to be characterized by a family of functions f_γ(x) depending on a certain parameter γ. Usually, we are, however, not able to foresee which of these functions shall be observed. Hence, in modelling of majority of real phenomena function f(·) have to be replaced by a random function f_γ(·); parameter γ is then interpreted as element of the space of elementary events Γ on which a probability is defined.