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

Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics

alt="script upper Y prime"/>, are the same function if and only if

script upper Y left-parenthesis omega right-parenthesis equals script upper Y Superscript prime Baseline prime left-parenthesis omega right-parenthesis for each omega element-of normal upper Omega period

Does the definition of stochastic integral in I1, I2, I3 yield such a unique value for integral Subscript 0 Superscript t Baseline script upper Z left-parenthesis s right-parenthesis d script upper X left-parenthesis s right-parenthesis ? I2 and I3 do not guarantee uniqueness: there may be different sequences left-brace script upper Z Superscript left-parenthesis p right-parenthesis Baseline right-brace in I2 which converge “in mean square” to . In effect, I4 asserts weak convergence of the integrals script upper S Subscript t Superscript left-parenthesis p right-parenthesis of the step functions to a value for the integral of script upper Z , that value being not necessarily unique.

If the integral does not have a unique value, what connections may exist between alternative values? Suppose there is more than one candidate random variable, say script upper Y and script upper Y prime , for the value of the stochastic integral,

script upper Y 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 script upper Y Superscript prime Baseline prime equals integral Subscript 0 Superscript t Baseline script upper Z left-parenthesis s right-parenthesis d script upper X left-parenthesis s right-parenthesis

In that case, what is the relation between script upper Y and script upper Y prime ? For instance, is it the case that, for each real number a, the probabilities of corresponding measurable sets are equal (such as upper P left-parenthesis script upper Y not-equals a right-parenthesis equals upper P left-parenthesis script upper Y prime not-equals a right-parenthesis ):

upper P left-parenthesis script upper Y less-than a right-parenthesis equals upper P left-parenthesis script upper Y Superscript prime Baseline prime less-than a right-parenthesis comma upper P left-parenthesis script upper Y greater-than a right-parenthesis equals upper P left-parenthesis script upper Y Superscript prime Baseline prime greater-than a right-parenthesis question-mark

The framework outlined above does not include the important case integral Subscript 0 Superscript t Baseline d script upper X left-parenthesis s right-parenthesis squared equals t , where left-parenthesis script upper X left-parenthesis s right-parenthesis right-parenthesis ( 0 less-than s less-than-or-equal-to t ) is Brownian motion. Broadly speaking, means that the random variables represented by finite sums

sigma-summation Underscript j equals 1 Overscript n Endscripts left-parenthesis script upper X left-parenthesis t Subscript j Baseline right-parenthesis minus script upper X left-parenthesis t Subscript j minus 1 Baseline right-parenthesis right-parenthesis squared

converge as t Subscript j Baseline minus t Subscript j minus 1 tend to zero, each j. In fact the convergence is weak, not point‐wise, with

normal upper E left-parenthesis sigma-summation Underscript j equals 1 Overscript n Endscripts left-parenthesis script upper X left-parenthesis t Subscript j Baseline right-parenthesis minus script upper X left-parenthesis t Subscript j minus 1 Baseline right-parenthesis squared right-parenthesis right-arrow t comma

and the weak limit t is a fixed real number which can be regarded as a degenerate random variable. This result is basic to the construction I1, I2, I3, I4.

A closer reading of source material may provide answers and/or corrections to some or all of the above comments and queries. Any misinterpretation, confusion, and errors may be dispelled by closer examination of the underlying ideas.

Aside from these issues, and looking beyond the classical mathematical theory, the general idea of stochastic integral is, in intuitive terms, a persuasive, natural and practical way of thinking about the underlying reality.

An alternative (and hopefully more understandable) mathematical way of representing this reality is presented in subsequent chapters of this book.

Example 2

In order to focus on the underlying ideas, here is a simple illustration. Suppose, at different times t, upper Y Subscript t (or upper Y left-parenthesis t right-parenthesis ) is a random variable, with sample space normal upper Omega . For simplicity suppose, for each t, has the same sample space and the same probability distribution. Suppose further that has only a finite number m of possible values y Subscript t Baseline 1 Baseline comma ellipsis comma y Subscript t m Baseline , each equally likely. Then we can take

normal upper Omega equals StartSet 1 comma 2 comma ellipsis comma m EndSet comma

and probability upper P left-parenthesis upper Y Subscript t Baseline left-parenthesis omega right-parenthesis equals i right-parenthesis equals StartFraction 1 Over m EndFraction . For upper A subset-of-or-equal-to normal upper Omega ,

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