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

trivially. In effect, the random‐variable‐as‐measurable‐function approach of classical theory reduces to the “naive” or “realistic” method, in which the probabilities pertain to outcomes , and are not primarily inherited from some abstract measurable space .

Alternatively, let the sample space be and let be the class of Borel subsets of (so includes the singletons for each ). Define on by and

so is atomic. As before, with ,

Classical probability involves a quite heavy burden of sophisticated and complicated measure theory. There are good historical reasons for this, and it is unwise to gloss over it. In practice, however, the sample space , in which probability measure is specified, is often chosen—as above—in such a way that measure‐theoretic abstractions and complexities melt away, so that the “natural” or untutored approach, involving just outcomes and their probabilities, is applicable.

[MTRV] shows how to formulate an effective theory of probability 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 5above and in the ensuing discussion, and which is relevant in stochastic integration.

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 5the 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