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

, and the proportion of weights in each interval is denoted by . A representative weight is chosen from each interval . The function is 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

      while the variance of the weights is approximately

      The latter calculation, involving , has the form with . The expressions and have the form of Riemann sums, in which the interval of real numbers is partitioned by the intervals , and where each is a representative data‐value in the corresponding interval . Thus the sums

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

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

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 is time, measured in days. Suppose a share, or unit of stock, has value on day ; suppose is the number of shares held on day ; and suppose 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 be the cumulative change in shareholding value at end of day , so . If share value and stockholding are subject to random variability, how is the gain (or loss) from the stockholding to be estimated?

      Take initial value (at time ) of the share to be (or ), take the initial shareholding or number of shares owned to be (or ). Then, at end of day 1 (),

      (2.1)