Stochastic Differential Equations

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.

In the observation of reality we often encounter phenomena which cannot be uniquely characterized. Dealing with such phenomena we observe that under the same conditions, the outcomes of a repeated experiment do not coincide; between the results there exist some unpredictable, random variations. Such phenomena are commonly regarded as random phenomena. Of course, the concept of “randomness” comes from our experience with a real world and is mostly used in its intuitive sense. Most often an event is regarded to be “random” if we do not know all causes and conditions of its realization.

As in the classical mathematical analysis in the theory of stochastic processes we come naturally to concepts such as: continuity, derivatives and integrals of a stochastic process (or, the Stieltjes-type integrals with respect to a stochastic process). Of course, these notions are of basic importance in the formulation and analysis of stochastic differential equations. It is, therefore, essential to define them precisely and to make clear the mutual relationships between various possible interpretations.

The basic theoretical problems concerned with stochastic differential equations are, generally speaking, the same as those in the case of deterministic differential equations, namely: existence and uniqueness of a solution, analytical properties of the solutions, dependence of the solutions on parameters and initial values etc. Yet, the introduction of random elements into appropriate differential equations leads to new probabilistic problems and specific difficulties. For instance, the sense itself of a stochastic differential equation should be clearly defined since it can be different depending on the understanding of a stochastic process and its derivatives. For the analysis of stochastic differential equations, however, a crucial point is regularity of random functions occuring in a given equation.

In addition to the theory of stochastic differential equations (dealing with such qualitative problems as these presented in the previous chapter) there exists now a large body of research devoted to the development of the effective methods of obtaining solutions. By the term “solution” we understand a stochastic process satisfying the equation together with its probabilistic properties.

The importance of differential equations in mathematics itself and in studying phenomena of the real world can not be over-estimated. The differential models of real phenomena constitute the only form which completely satisfies the modern physicist’s demand for causality. A. Einstein (1930) pointed out that “the clear conception of the differential law is one of Newton’s greatest achievements”. And R. Thom (1973) went even much further saying that: “the possibility of using the differential model is, to my mind, the final justification for the use of quantitative models in the sciences”.