Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics. Patrick Muldowney

book [70] has the title Integration in Statistics. It deals mostly with tests of significance, and touches on some questions of probability theory using the Riemann‐complete method. The 1955 paper is concerned strictly with the nature of integration. But ancillary matters such as probability—and, indeed, differentiation—featured consistently in Henstock's subsequent work.

      5 ⁶ In the terminology of [MTRV] and this book, Feynman's method consists of substituting cylinder function approximations in the action functional.

      6 ⁷ This simplification represents each of the variables , , and as one‐dimensional. The electric field component is essentially vectorial, and one‐dimensional is contrary to the physical nature of the system. A physically more accurate version can be arrived at by a careful reading of chapter 9 of [FH]. And even though it is a bit more complicated, it is not too difficult to adapt the mathematical theory presented in this book.

      7 ⁸ A Cauchy sum has . But allowing to be either of or makes a connection with the Riemann sums of ‐complete integration.

      8 ⁹ Theorem 63 (page 175 of [MTRV]) is false. See Section 11.2 below; and also [ website ].

Part I Stochastic Calculus

Chapter 1 Stochastic Integration

      The idea or purpose of stochastic integration is to define a random variable

      where is a random or unpredictable quantity, depending in a particular manner on unpredictable entities and ; and where

      are stochastic processes and depends on time t. In textbooks, the integrand is usually presented as , but is used here in order to emphasise that the integrand is intended to be random.

      The integrand (or, when appropriate, ) is to be regarded as a measurable function—as is —with respect to a probability space .

      If is a deterministic or non‐random function of s, its value at time s is a definite (non‐random) number which, whenever necessary, can be regarded as a degenerate random variable. If is the same random variable for each s in , each j, then the process is a step function. (In textbooks, the term elementary function is often applied to this.)

      The most important kind of stochastic integral is where , is standard Brownian motion, and this particular case (called the Itô integral) is outlined here. The main steps are as follows.

       I1 Suppose the integrand is a step function, with constant random variable value for , . Then defineIn this case (that is, a step function), the Itô isometry holds for expected values:

       I2 Suppose the process (not necessarily a step function) satisfiesThen there exists a sequence of step functions (processes) , such that

       I3 For such , define its stochastic integral with respect to the process as

       I4 If is Brownian motion the latter limit exists.

      An objective of this book is to provide an alternative to the classical theory, not develop it. Thus the commentary, interpretation, and speculation of this section can be safely omitted by anybody who is either already familiar with, or is not interested in, the standard theory of stochastic integration.

      Regarding notation, many textbooks use the symbol B for Brownian motion, whereas is used above. Textbooks also use the symbol for the integrand, where is used above. The reason for using notation instead of ) is to emphasise that the value of the integrand function is generally a random variable depending on s, and not generally a single, definite real or complex number (such as the deterministic function , for instance) of the kind which occurs in ordinary integration.

In classical probability theory, an underlying mathematical probability measure space Скачать книгу