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

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 a “naive” or “realistic” way—the stochastic integral of the process with respect to the process . In chapter 8of [MTRV], symbols , or , or are used (in place of the symbol ) for various kinds of stochastic integral. In the context described here, would be the appropriate notation. (See (5.28) below.)

To illustrate the details of this basic stochastic integration, suppose the time increment is 1 day, so tracks the process over four days. Suppose the initial value of the share at the start of day 1 is . Suppose on each day the value of the share can change by or . That is, an “up” increment (U) or “down” increment (D). (Although the probabilities involved will not be used at this stage, in order to keep random variability in mind suppose that, at the end of each day, U occurs with probability and suppose D occurs with probability .)

Suppose initial stockholding at start of day 1 is , or 1 share, and suppose the shareholder buys an extra share whenever the share value increases (U), and otherwise keeps the same number of shares. So there are no circumstances in which shareholding is decreased. (It is easy to imagine that the investor would apply a less optimistic and more prudent share purchasing strategy. But for purpose of illustration some particular strategy must be chosen, and this one is easy to describe.)

The up (U) or down (D) changes in share price over four days are listed in Table 2.4. There are , , possible processes or histories, corresponding to the 16 possible permutations‐with‐repetition of the 2 symbols U and D, taken four at a time.

With and , the histories or processes of interest are prices ; holdings ; and total gains ; represented by