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

members, one of which (for example) is

with consisting of five individual four‐tuples. Assume that Up transitions and Down transitions are equally likely, and that they are independent events. Then, as before,

      for each . For above,

      To relate this probability structure to the shareholding example, let , and let

      (2.14)

      using Table 2.4; so, for instance,

      and so on. Next, let denote the stochastic integrals of the preceding section, so for ,

      so gives the values of Table 2.4. As described in Section 2.3, the rationale for deducing the probabilities of outcomes , , from the probabilities on is the relationship

      For this calculation to work in general, the functions involved ( and ) must be measurable. But that is no problem in this case since all the sets involved are finite. To illustrate the calculation, take . Then, referring to Table 2.4 whenever necessary,

      As a further illustration, suppose we now take

(2.15)

      letting be the sigma‐algebra of Borel subsets of . For each define as follows. Suppose and