The construction in I1, I2, I3 indicates an integration domain or . (There is nothing surprising in that.) But in I1, I2, I3 the integrand values are not real or complex numbers, but random variables—which may be a bit surprising.
But it is not unprecedented. For instance, the Bochner integration process in mathematical analysis deals with integrands whose values are functions, not numbers.
The construction and definition of the Bochner integral [105] is similar in some respects to the classical Itô integral. What is the end result of the construction in I1, I2, I3?
In general, the integral of a function f gives a kind of average or aggregation of all the possible values of f. So if each value of the integrand is a random variable, the integral of f should itself be a random variable—that is, a function which is measurable with respect to an underlying probability measure space.
If the notation is valid or justifiable for the stochastic integral, it suggests that the Itô integral construction derives a single random variable (or ) from many jointly varying random variables, such as , as varies between the values 0 and t. This is reminiscent of Norbert Wiener's construction in [169], which is in some sense a mathematical replication in one dimension of Brownian motion; even though the latter is essentially an infinite‐dimensional phenomenon with infinitely many variables. Without losing any essential information, a situation involving infinitely many variables is converted to a scenario involving only one variable.3
The proof of the Itô isometry relation (see I1) indicates that, as a stochastic process, must be independent of . Otherwise the construction I1, I2, I3 would seem to be inadequate as it stands, whenever the process is replaced by a process .
In I3 the integrand does not have step function form; and, on the face of it, indicates dependence of (or ) on random variables and for everys, . If the integrand were (which, in general, it is not), with joint random variability for , and if is Brownian motion, then the joint probability space for the processes and is given by the Wiener probability measure and its associated multi‐dimensional measure space. (The latter are described in Chapter 5 below.)
Returning to I1, the Itô integral of step function is defined as
where the are random variable values of . It is perfectly valid to combine finite numbers of random variables in this way, in order to produce, as outcome, a single random variable (—which may be a joint random variable depending on many underlying random variables).
This part of the formulation of the integral of a step function in I1 corresponds to the integral of a step function in basic integration, and does not require any passage to a limit of random variables.
not approximate or estimated).
