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

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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

      The idea or purpose of stochastic integration is to define a random variable script upper S Subscript t Baseline equals

integral Subscript 0 Superscript t Baseline script upper Z left-parenthesis s right-parenthesis d script upper X left-parenthesis s right-parenthesis comma or integral Subscript 0 Superscript t Baseline f left-parenthesis script upper X left-parenthesis s right-parenthesis right-parenthesis d script upper X left-parenthesis s right-parenthesis

      where script upper S Subscript t is a random or unpredictable quantity, depending in a particular manner on unpredictable entities script upper X and script upper Z; and where

script upper X comma equals left-parenthesis script upper X left-parenthesis s right-parenthesis colon 0 less-than s less-than-or-equal-to t right-parenthesis comma script upper Z comma equals left-parenthesis script upper Z left-parenthesis s right-parenthesis colon 0 less-than s less-than-or-equal-to t right-parenthesis

      are stochastic processes and script upper S Subscript t depends on time t. In textbooks, the integrand is usually presented as f left-parenthesis s right-parenthesis, but script upper Z left-parenthesis s right-parenthesis is used here in order to emphasise that the integrand is intended to be random.

      The integrand script upper Z left-parenthesis s right-parenthesis (or, when appropriate, f left-parenthesis script upper X left-parenthesis s right-parenthesis right-parenthesis) is to be regarded as a measurable function—as is script upper X left-parenthesis s right-parenthesis—with respect to a probability space left-parenthesis normal upper Omega comma script upper A comma upper P right-parenthesis.

      If script upper Z left-parenthesis s right-parenthesis is a deterministic or non‐random function g left-parenthesis s right-parenthesis 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 script upper Z left-parenthesis s right-parenthesis is the same random variable for each s in t Subscript j minus 1 Baseline less-than-or-equal-to s less-than t Subscript j, each j, then the process script upper Z is a step function. (In textbooks, the term elementary function is often applied to this.)

      The most important kind of stochastic integral is where script upper X comma equals left-parenthesis script upper X left-parenthesis s right-parenthesis right-parenthesis Subscript 0 less-than s less-than-or-equal-to t Baseline, 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 script upper X is used above. Textbooks also use the symbol f left-parenthesis s right-parenthesis for the integrand, where script upper Z left-parenthesis s right-parenthesis is used above. The reason for using notation script upper Z left-parenthesis s right-parenthesis instead of f left-parenthesis s right-parenthesis) 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 g left-parenthesis s right-parenthesis equals s squared, for instance) of the kind which occurs in ordinary integration.