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

which follows naturally from the naive or realistic approach described above, and which does not require the theory of measure as its foundation. The following pages are intended to convey the basic ideas of this approach.

Before moving on to this, here is an elaboration of a technical point of a financial character, which appeared in Example 5 above and in the ensuing discussion, and which is relevant in stochastic integration.

      Example 6

Expression (2.5) above gives two representations of a stochastic integral,

      based on sample value calculations (2.4:

      (2.10)

derived from (2.2) and (2.3):

      If and are to be treated as functions of a continuous variable for , this suggests calculations or estimates on the lines of

(2.11)

      where is a partition of .

      For Example 5 the sample calculation (2.4) of total portfolio value leads unproblematically to the random variable representation (2.5), . Though we have not yet settled on a meaning for stochastic integral, the discrete expression

      points towards as a continuous variable form of stochastic integral. It seems that the sample value form of the latter should be the Riemann‐Stieltjes integral , for which a Riemann sum estimate is

(2.12)

      where for .

But (2.11) has , not the of (2.12). The logic of Example 5 indicates that only the left hand value is permitted in the Riemann sum estimates of the stochastic integral . Why is this?

      The issue is to choose between two forms of Riemann sum: