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

Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics

rel="nofollow" href="#fb3_img_img_b5bc439e-68a5-506e-9457-f0f82fde846f.png" alt="normal upper Omega"/> for this experiment consists of the pair of numbers,

and

. Let

be the family of all subsets of normal upper Omega

normal upper Omega equals StartSet 0 comma 1 EndSet comma script upper A equals StartSet empty-set comma StartSet 0 EndSet comma StartSet 1 EndSet comma StartSet 0 comma 1 EndSet EndSet semicolon

and define a probability measure upper P by

upper P left-parenthesis empty-set right-parenthesis equals 0 comma upper P left-parenthesis StartSet 0 EndSet right-parenthesis equals one half comma upper P left-parenthesis StartSet 1 EndSet right-parenthesis equals one half comma upper P left-parenthesis StartSet 0 comma 1 EndSet right-parenthesis equals upper P left-parenthesis normal upper Omega right-parenthesis equals 1 period

Then, trivially, script upper A is a sigma ‐algebra of subsets of normal upper Omega , and upper P is, trivially, countably additive¹ on , so left-parenthesis normal upper Omega comma script upper A comma upper P right-parenthesis is a probability measure space.

The set of outcomes of a single throw of a coin is the set upper V equals StartSet normal upper H comma normal upper T EndSet , and the family of subsets of upper V is

script upper V equals StartSet empty-set comma StartSet normal upper H EndSet comma StartSet normal upper T EndSet comma StartSet normal upper H comma normal upper T EndSet EndSet semicolon

and left-parenthesis upper V comma script upper V right-parenthesis is a measurable space. Define the following function to represent the coin tossing experiment:

script upper X colon normal upper Omega right-arrow from bar upper V comma script upper X left-parenthesis 0 right-parenthesis equals normal upper H comma script upper X left-parenthesis 1 right-parenthesis equals normal upper T period

Then, for upper S element-of script upper V , script upper X Superscript negative 1 Baseline left-parenthesis upper S right-parenthesis element-of script upper A so is a upper P ‐measurable function.

There are many different ways of defining the probability space. It is natural to use real‐number‐valued functions, so the outcomes upper V can be denoted and instead of H and T, and the measurable function script upper X can then be the identity function. (It is usual that the values of a random variable are represented as real numbers² ; with expected—mean or average—value, variance, and so on; which are also real numbers.)

But no matter what way this construction is done, the classical, rigorous mathematical representation by measurable function is evidently more complicated than the naive or natural view of the coin tossing experiment. In contrast, the purpose of this book is to provide a rigorous theory of stochastic integration/summation which (like [MTRV]) bypasses the “measurable function” view, and which is closer to the “naive realistic” view.

Example 3

Throw a pair of dice and, whenever the sum of the numbers observed exceeds 10, pay out a wager equal to the sum of the two numbers thrown, and otherwise receive a payment equal to the smaller of the two numbers observed. If the two are the same number (with sum not exceeding 10) then the payout is that number.

In Example 3 take sample space

StartLayout 1st Row 1st Column normal upper Omega 2nd Column equals 3rd Column StartSet 1 comma 2 comma 3 comma 4 comma 5 comma 6 EndSet times StartSet 1 comma 2 comma 3 comma 4 comma 5 comma 6 EndSet comma with 2nd Row 1st Column script upper A 2nd Column equals 3rd Column 2 Superscript normal upper Omega Baseline left-parenthesis the set of all subsets of normal upper Omega right-parenthesis comma 3rd Row 1st Column upper P left-parenthesis upper A right-parenthesis 2nd Column equals 3rd Column StartFraction StartAbsoluteValue upper A EndAbsoluteValue Over StartAbsoluteValue normal upper Omega EndAbsoluteValue EndFraction left-parenthesis the cardinality of upper A divided by the cardinality of normal upper Omega right-parenthesis period EndLayout

Observation of a throw of the pair of dice can be represented by a listing of the possible joint outcomes script upper X equals left-brace left-parenthesis x 1 comma x 2 right-parenthesis right-brace , x 1 equals 1 comma 2 comma ellipsis comma 6 , x 2 equals 1 comma 2 comma ellipsis comma 6 . Define a random variable g left-parenthesis script upper X right-parenthesis colon normal upper Omega right-arrow from bar bold upper R by

g left-parenthesis x 1 comma x 2 right-parenthesis equals x 1 plus x 2

for each left-parenthesis x 1 comma x 2 right-parenthesis element-of normal upper Omega . Then, as in the previous example where the domain and range of script upper X are finite sets, g left-parenthesis script upper X right-parenthesis

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