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

and , respectively, ensures measurability in and t. It also ensures measurability for the conditional cases of (or ) with already determined as known real numbers when .

      Let represent sample values (or potential occurrences) of the random variables . For any given j, i can have value

      so is the set of permutations, with repetition, of the numbers taken n at a time.

      The ideas in I1, I2, I3, I4) suggest the following sample values for the stochastic integral (or “”):

      The subscript labels the random variability in this calculation, and demonstrates that this version of the stochastic integral can take possible values; though not all of the possible values are necessarily distinct.

      For further simplification, take and ; so, at each of times (), the random variable can take one of two possible values, . Then, by enumerating the permutations with repetition of things taken at a time , the 8 possible sample values of the stochastic integral are:

      Now suppose that the deterministic function f is exponentiation to the power of 2 (so ); and suppose the random variable (or above) has sample values and with equal probabilities . Calculating each of the above expressions, the 8 sample evaluations of the stochastic integral are, respectively,