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

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target="_blank" rel="nofollow" href="#ulink_677b49e4-e938-59db-ac24-00f21ae0bbbb">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)

      At end of day ,

(2.2)

      After days,

(2.3)

      If the time increments are reduced to arbitrarily small size (so represents number of “time ticks”—fractions of a second, say), with the meaning of the other variables adjusted accordingly, then

(2.4)

      The latter expressions are Riemann sum estimates of (a Stieltjes‐type integral) whenever the latter exists.

Each of the expressions in (2.4) is sample value of a random variable

(2.5)

constructed from the random variables , , and . These notations symbolize—in

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