A Modern Theory of Random Variation: With Applications in Stochastic Calculus, Financial Mathematics, and Feynman Integration

Chapter 9

Numerical Calculation

A theme of this book is that Riemann sums are an effective means of analyzing random variability phenomena, enabling a comprehensive theory to be constructed. The traditional theory of probability is based on measurable sets formed by operations involving infinitely many steps. In contrast, Riemann sums have only a finite number of terms. On the face of it, such sums should be relatively easy to calculate. This chapter contains a number of such calculations, using Maple 15 computer software.

Numerical and empirical investigations warrant detailed study in their own right. But such an project is beyond the scope of this book. All that can be given are a few indications and pointers.

In preceding chapters Riemann sums have been used to calculate and analyze measurability of sets and functions, expectation values of random variables, state functions of diffusion systems and quantum mechanics, Feynman diagrams, valuation of share options, and strong and weak stochastic integrals.

In order to produce a robust theory, considerable subtlety has been built into the construction of Riemann sums. By making the Riemann sums conform to construction rules called gauges it has been possible to establish rules and criteria for sophisticated mathematical operations by which many classical results could be deduced, and also many new results which are beyond the reach of traditional theory.

But the essential simplicity of Riemann sums is a constant feature. Furthermore, ...