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

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 notindependent, 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)

whose elements are the different values which can be taken by the variable . For instance, , which is a member of the family of all subsets of .

Following through the logic of the classical theory, probability is defined on the family of measurable subsets of . A random variable is a real‐number‐valued, and ‐measurable, function

in this case, where the potential values are the numbers in the right‐most column of Table 2.4. The latter set is finite; and every finite subset, such as , is measurable. In fact, with sample space chosen in this way, is the identity function, since we have chosen so that its elements are the distinct values .

To find the probability of a set of ‐outcomes, such as , the classical theory requires that the corresponding set be found so that

gives the probability of outcomes as the corresponding probability in the sample space. Conveniently, in this case is chosen as simply the set of outcomes ; is the identity function; and