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

Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics

left-parenthesis normal upper Omega comma script upper A comma upper P right-parenthesis

‐measurable and qualifies as a random variable, with expected value

normal upper E left-parenthesis g left-parenthesis script upper X right-parenthesis right-parenthesis equals integral Underscript normal upper Omega Endscripts g left-parenthesis script upper X right-parenthesis d upper P equals 7 period

The integral in this case reduces to the sum of a finite number of terms.

The payoff from the wager in Example 3 is a randomly variable amount given by

script upper Z equals f left-parenthesis script upper X right-parenthesis comma script upper Z left-parenthesis x 1 plus x 2 right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column minus left-parenthesis x 1 plus x 2 right-parenthesis 2nd Column if 3rd Column x 1 plus x 2 4th Column greater-than-or-equal-to 5th Column 10 comma 2nd Row 1st Column x 1 2nd Column if 3rd Column x 1 4th Column less-than-or-equal-to 5th Column x 2 comma 3rd Row 1st Column x 2 2nd Column if 3rd Column x 2 4th Column less-than-or-equal-to 5th Column x 1 period EndLayout

In this case, script upper Z is a composite of the deterministic function with the random variable script upper X ; and, just like g left-parenthesis script upper X right-parenthesis , is (trivially) left-parenthesis normal upper Omega comma script upper A comma upper P right-parenthesis ‐measurable, and is a random variable, with

normal upper E left-parenthesis script upper Z right-parenthesis equals normal upper E left-parenthesis f left-parenthesis script upper X right-parenthesis right-parenthesis equals integral Underscript normal upper Omega Endscripts script upper Z left-parenthesis omega right-parenthesis d upper P equals StartFraction 41 Over 36 EndFraction semicolon

where, again, the Lebesgue integral reduces (trivially) to a finite sum of terms.

There are many alternative ways of representing mathematically the unpredictable payout of this wager, as the following illustration shows. The outcome of the wager is the value , where

y equals StartLayout Enlarged left-brace 1st Row 1st Column minus left-parenthesis x 1 plus x 2 right-parenthesis 2nd Column if 3rd Column x 1 plus x 2 4th Column greater-than-or-equal-to 5th Column 10 comma 2nd Row 1st Column x 1 2nd Column if 3rd Column x 1 4th Column less-than-or-equal-to 5th Column x 2 comma 3rd Row 1st Column x 2 2nd Column if 3rd Column x 2 4th Column less-than-or-equal-to 5th Column x 1 period EndLayout

Examining each of the 36 pairs in left-brace left-parenthesis x 1 comma x 2 right-parenthesis right-brace ( 1 less-than-or-equal-to x 1 less-than-or-equal-to 6 , 1 less-than-or-equal-to x 2 less-than-or-equal-to 6 ) in turn, the outcomes or payouts are listed in Table 2.1.

For instance, of the 36 possible pairs of throws left-parenthesis x 1 comma x 2 right-parenthesis , a loss of 11 is incurred twice, with throws of (5,6) and (6,5).

Accordingly, let the sample space for the wager be

normal upper Omega prime equals StartSet 1 comma 2 comma 3 comma 4 comma 5 comma negative 11 comma negative 12 EndSet comma

let the measurable sets script upper A prime consist of 2 Superscript normal upper Omega prime , the family of all subsets of , and, for y element-of normal upper Omega prime let upper P left-parenthesis StartSet y EndSet right-parenthesis be as set out in Table 2.1; so, for upper A element-of script upper A prime , upper P left-parenthesis upper A right-parenthesis can be found by adding up the relevant probabilities in Table 2.1.

Table 2.1 Distribution of payouts.

Payout	Probability
1
2
3
4
5
‐11
‐12

Now define random variable script upper Y colon normal upper Omega prime right-arrow from bar bold upper R by the identity mapping script upper Y left-parenthesis omega prime right-parenthesis equals y for Скачать книгу