Stochastic Integration and Differential Equations— Physical Approach

Abstract

The theory of integrals of deterministic functions and solutions of differential equations, from the point of view of real analysis, is based on the fundamental notion of the limit of a sequence. With the advent of measure theory, Lebesgue integration, and generalized functions it has become possible to extend the idea of integration to the widest class of deterministic functions. It is therefore a reasonable question to ask whether random functions can be treated in a similar fashion. In fact, integrals of random functions which, for brevity, may be called stochastic integrals, arise from very many physical situations. Stochastic integrals are very well known to electrical engineers who very often have to deal with responses to random signals and noise. From a pure mathematician’s point of view, limiting stochastic operations do not offer any special difficulty since all limiting stochastic operations follow from the notion of convergence almost everywhere, convergence in mean square and convergence in measure, provided we replace Lebesgue measure by probability and the ordinary space by the function space of random functions. While this analogy is useful in that it provides a sound mathematical basis for the formulation of probability problems, it does not enable us to compute quantities of physical significance. In fact, the situation is analogous to the theory of Riemannian integration where the evaluation of integrals is made by the use of primitives or by the use of Simpson’s formula interpreting the integral as an area.