alt="upper P left-parenthesis upper Y Subscript t Baseline element-of upper A right-parenthesis equals StartFraction StartAbsoluteValue upper A EndAbsoluteValue Over m EndFraction comma"/>
where is the number of elements in A. Then, for each t, is a ‐measurable function and thus a random variable. (We may also suppose, if it is convenient for us, that for any t, , the random variables are independent.)
Now suppose that, for , is another indeterminate or unpredictable quantity; and that, for given t, the possible values of depend in some deterministic way on the corresponding values of , so
where f is a deterministic function. For instance, the deterministic relation could be , so if the value taken by at time t is , then the value that takes is . Provided f is a “reasonably nice” function (such as ), then is measurable with respect to , and is itself a random variable.
This scenario is in broad conformity with I1, I2, I3, I4 above. So it may be possible to consider, in those terms, the stochastic integral of with respect to . Essentially, with , then for each t, for , and for ,
the two formulations being equivalent. If the stochastic integral “” is to be formulated in terms of Lebesgue integrals in (as intimated in I1, I2, I3, I4), then some properties of t‐measurability () are suggested. This aspect can also be simplified, as follows.
Just as was reduced to a finite number m of possible values, 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
