can be replaced by a finite number of fixed time values if the family of random variables () is replaced by (); so there are only a finite number n of random variables ,
and the random variables can be written
for , . (Below, will be taken to be .) Replacing the domains and by and , respectively, ensures measurability in and t. It also ensures measurability for the conditional cases of (or ) with already determined as known real numbers when .
Let represent sample values (or potential occurrences) of the random variables . For any given j, i can have value
so is the set of permutations, with repetition, of the numbers taken n at a time.
The ideas in I1, I2, I3, I4) suggest the following sample values for the stochastic integral (or “”):
The subscript labels the random variability in this calculation, and demonstrates that this version of the stochastic integral can take possible values; though not all of the possible values are necessarily distinct.
For further simplification, take and ; so, at each of times (), the random variable can take one of two possible values, . Then, by enumerating the permutations with repetition ofthings takenat a time , the 8 possible sample values of the stochastic integral are:
