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 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
alt="s minus 1"/> is
