Modern probability theory

Abstract

Introduction You are probably asking yourself why we need a second chapter on probability theory. The reason is that the modern formalism used by mathematicians to describe probability involves a number of concepts, predefined structures, and jargon that are not included in the simple approach to probability used by the majority of natural scientists, and the approach we have adopted here. This modern formalism is not required to understand probability theory. Further, in the author's experience, the vast majority of physically relevant questions can be answered, albeit nonrigorously, without the use of modern probability theory. Nevertheless, research work that is written in this modern language is not accessible unless you know the jargon. The modern formalism is used by mathematicians, some mathematical physicists, control theorists, and researchers who work in mathematical finance. Since work that is published in these fields is sometimes useful to physicists and other natural scientists, it can be worthwhile to know the jargon and the concepts that underly it. Unfortunately a considerable investment of effort is required to learn modern probability theory in its technical detail: significant groundwork is required to define the concepts with the precision demanded by mathematicians. Here we present the concepts and jargon of modern probability theory without the rigorous mathematical technicalities. Knowing this jargon allows one to understand research articles that apply this formalism to problems in the natural sciences, control theory, and mathematical finance.