Patrick Muldowney - Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics

Here, use of the symbol for the integrand (instead of the usual ) indicates that while the integrand is a random variable dependent on s , it does not necessarily depend on the integrator random variable . If, in fact, there is such dependence, then an appropriate notation 2 for the integrand is .

The notation and terminology of ordinary integration is used in I1, I2, I3, I4, and they provide a certain “feel” for what is going on. But the various elements of the system are clearly different from ordinary integration. Can we get some more precise idea of what is really going on?

The “integration‐like” construction in I1suggests that the domain of integration is , and that the integrand takes values in a class of functions (—random variables; that is, functions which are measurable with respect to some probability space, or spaces).

How does this compare with more familiar integration scenarios? Basic integration (“ ”) has two elements: firstly, a domain of integration containing values of the integration variable s , and secondly, an integrand function which depends on the values s in the domain of integration. The more familiar integrand functions have values which are real or complex numbers ; and which are deterministic (that is, “definite”, not approximate or estimated).

The construction in I1, I2, I3indicates an integration domain or . (There is nothing surprising in that.) But in I1, I2, I3the 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, I3would seem to be inadequate as it stands, whenever the process is replaced by a process .

In I3the integrand does not have step function form; and, on the face of it, indicates dependence of (or ) on random variables and for every s , . 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 5below.)