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

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Название Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics
Автор произведения Patrick Muldowney
Жанр Математика
Серия
Издательство Математика
Год выпуска 0
isbn 9781119595526



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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 x is chosen from each interval upper I. The function f left-parenthesis x right-parenthesis is x squared 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 f left-parenthesis x right-parenthesis equals x squared. The expressions 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 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 x 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 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.

      The following is similar to Example 2.

      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 s; suppose z left-parenthesis s right-parenthesis is the number of shares held on day s; and suppose c left-parenthesis s right-parenthesis is the change in the value of the shareholding on day s as a result of the change in share value from the previous day so c left-parenthesis s right-parenthesis equals z left-parenthesis s minus 1 right-parenthesis left-parenthesis x left-parenthesis s right-parenthesis minus x left-parenthesis s minus 1 right-parenthesis right-parenthesis. Let w left-parenthesis s right-parenthesis be the cumulative change in shareholding value at end of day s, so w left-parenthesis s right-parenthesis equals w left-parenthesis s minus 1 right-parenthesis plus c left-parenthesis s right-parenthesis. If share value x left-parenthesis s right-parenthesisand stockholding z left-parenthesis s right-parenthesisare 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),

      (2.1)c left-parenthesis 1 right-parenthesis equals z left-parenthesis 0 right-parenthesis times left-parenthesis x left-parenthesis 1 right-parenthesis minus x left-parenthesis 0 right-parenthesis right-parenthesis comma w left-parenthesis 1 right-parenthesis 
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