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

      the sum being taken over all 16 values (including duplicate values) of total gain .

      When duplicate values are combined, there are 12 distinct outcomes for . Each of the duplicated outcomes has probability , while each of the other 8 distinct outcomes has probability .

      To find the expected value of (or ) in accordance with the classical, rigorous mathematical theory of probability, it should be formulated in terms of a probability space , so

(2.7)

      There are many ways in which a sample space can be constructed. One way is to let be the set of numbers consisting of the different values of (i.e. without duplicate values), of which there are 12, and let be the appropriate atomic probability measure on these 12 values. Letting be the identity function on , (or ) is measurable (trivially), and

because the integral in (2.7) reduces to the sum in (2.6.

      Now suppose that, at times , the probability of an Up transition in is , while the probability of a Down transition in is :

      and suppose, as before, that Up or Down transitions are independent of each other; so, for instance, the joint transition sequence U‐D‐D‐U (and the corresponding ) has probability

      with similar probability calculations for each of the other 15 transition paths and their corresponding values (including duplicates, such as D‐U‐U‐U which also gives ).

      The 16 outcomes (including replicated outcomes) for accumulated gain are the same as before, but because the probabilities are different, the expected net gain is now

      (2.8)

      Both calculations reduce to the same finite sum of terms. It is seen here that the new probability distribution, favouring Up transitions, produces an overall net gain in wealth through the policy of acquiring shares on an up‐tick, while not shedding shares on a down‐tick—the “optimistic” policy, in other words.

      If the joint transition probabilities

      are not independent, then, provided the dependencies between the various transitions and events are known, it is still possible to calculate all the relevant joint probabilities. But generally this is not so simple as the rule (of multiplying the component probabilities) that obtains when the joint occurrences are independent of each other.

      A key step in the analysis is the construction of the probabilities for the values of the random variable (or ). The framework for this is as follows. Consider any subset of the sample space

      (2.9)