minus x left-parenthesis s Subscript j minus 1 Baseline right-parenthesis right-parenthesis period"/>
The latter corresponds to the calculation
of Example 5, where is used, but not or any value intermediate between and .
The reasoning is as follows. At time the investor makes a policy decision to purchase a quantity of shares whose value from time up to (but not including) time is . This number of shares (the portfolio) is retained up to time . At that instant of time the decision cycle is repeated, and the investor adjusts the portfolio by taking a position of holding number of shares, each of which has the new value .
In the time period to , the gain in value of the portfolio level chosen at time is
not , since the portfolio quantity operates in the time period to (not to ). Reverting to continuous form, this translates to Riemann sum terms of the form
2.4 Choosing a Sample Space
It was mentioned earlier that there are many alternative ways of producing a sample space (along with the linked probability measure and family of measurable subsets of ). The set of numbers
was used as sample space for the random variability in the preceding example of stochastic integration. The measurable space was the family of all subsets of , and the example was illustrated by means of two distinct probability measures , one of which was based on Up and Down transitions being equally likely, where for the other measure an Up transition was twice as likely as a Down.
An alternative sample space for this example of random variability is
(2.13)
where for ; so the elements of consist of sixteen 4‐tuples of the form
Let the measurable space be the family of all subsets of ; so contains
