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

Preface

The theory of probability is one of the success stories of twentieth century mathematics. Its success was founded on advances in the theory of integration associated with Henri Lebesgue, which, in turn, are based on the mathematical theory of measure.

But twentieth century probability theory is constrained by certain features of the Lebesgue integral. Lebesgue integration cannot safely be used without first mastering the underlying theory of measure—a subtle and difficult subject.

Furthermore, in Lebesgue integration, as in the Riemann integration that it superseded, a function is integrable only if its absolute value is integrable. Consequently, some perfectly straightforward functions cannot be integrated by Lebesgue’s method. This limitation meant that Richard Feynman’s mid-twentieth century discoveries in quantum mechanics, including the theory of light for which he received the Nobel Prize, could not be expressed in probability terms to which his theory bears a strong formal resemblance.

A further limitation is manifested in the Itô calculus used in financial mathematics and the theory of communication. This is because the Lebesgue version of stochastic calculus is relatively complicated and difficult to apply in practice.

This book overcomes these limitations by formulating probability theory in terms of the Stieltjes-complete integral instead of the Lebesgue integral. The roots of the Stieltjes-complete integral of this book are in developments in mathematical analysis ...