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_d1ef806f-6843-5c69-b5cf-54338edb9b26.png" alt="upper I"/>, and the proportion of weights in each interval upper I

is denoted by upper F left-parenthesis upper I right-parenthesis

. A representative weight

is chosen from each interval upper I

. The function f left-parenthesis x right-parenthesis

since, in this case, these values are needed in order to estimate the variance. Completing the calculation, the estimate of the arithmetic mean weight in the sample is

sigma-summation x upper F left-parenthesis upper I right-parenthesis equals 44 kg comma

while the variance of the weights is approximately

sigma-summation x squared upper F left-parenthesis upper I right-parenthesis minus left-parenthesis 44 right-parenthesis squared equals 2580 minus 1936 equals 644 kg squared period

The latter calculation, involving sigma-summation x squared upper F left-parenthesis upper I right-parenthesis , has the form sigma-summation f left-parenthesis x right-parenthesis upper F left-parenthesis upper I right-parenthesis with . The expressions sigma-summation x upper F left-parenthesis upper I right-parenthesis and have the form of Riemann sums, in which the interval of real numbers left-bracket 0 comma 100 right-bracket is partitioned by the intervals upper I , and where each is a representative data‐value in the corresponding interval upper I . Thus the sums

sigma-summation x upper F left-parenthesis upper I right-parenthesis and sigma-summation f left-parenthesis x right-parenthesis upper F left-parenthesis upper I right-parenthesis

are approximations to the Stieltjes (or Riemann–Stieltjes) integrals

integral Underscript upper J Endscripts x d upper F and integral Underscript upper J Endscripts f left-parenthesis x right-parenthesis d upper F comma respectively semicolon

the domain of integration [0,100] being denoted by upper J .

In Section 2.1 the variables are discrete. But the outcomes there can be expressed as discrete elements of a continuous domain provided the probabilities are formulated as atomic functions on the domain.

In contrast, the variables in Tables 2.2 and 2.3 are continuous, and their continuous domain is partitioned for Riemann sums in a natural way. Then Riemann sums can be formed as in Table 2.3.

2.3 A Basic Stochastic Integral

The following is similar to Example 2.

Example 5

Suppose s equals 1 comma 2 comma 3 comma ellipsis is time, measured in days. Suppose a share, or unit of stock, has value x left-parenthesis s right-parenthesis on day ; suppose z left-parenthesis s right-parenthesis is the number of shares held on day ; and suppose c left-parenthesis s right-parenthesis is the change in the value of the shareholding on day as a result of the change in share value from the previous day so . Let w left-parenthesis s right-parenthesis be the cumulative change in shareholding value at end of day , so . If share value x left-parenthesis s right-parenthesis and stockholding z left-parenthesis s right-parenthesis are subject to random variability, how is the gain (or loss) from the stockholding to be estimated?

Take initial value (at time s equals 0 ) of the share to be x left-parenthesis 0 right-parenthesis (or x 0 ), take the initial shareholding or number of shares owned to be z left-parenthesis 0 right-parenthesis (or z 0 ). Then, at end of day 1 ( s equals 1 ),

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