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

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 I1corresponds to the integral of a step function in basic integration, and does not require any passage to a limit of random variables.

Now suppose each is a fixed real number ; so, for , . (Accordingly, in I1, can be regarded as a “degenerate” random variable, with atomic probability value.) Suppose the integrator is the real‐valued ds instead of the random variable‐valued . Then 4

Formally, at least, this looks like the definition in I1of when is a step function. The factor equals for each j . This emerges naturally from the mathematical meaning of the length or distance variable s , and from the mathematical meaning of .

Can this be replicated in I1when is a step function, or when each is a fixed real number ? Is it the case that

With each , this would imply

(1.1)

If this is unproblematical, it should be possible to deduce it from one or other of the various mathematical definitions of Brownian motion , along with some mathematical definition of the integral in this context.

But it appears that there is no such understanding of . So, as in I1, it seems that this formulation is to be regarded as a basic postulate or axiom of stochastic integration.

Returning to the definition of the classical Itô integral, I2has the following condition on the expected value of the integral of the process :

The idea here is that, if is the random entity obtained by carrying out some form of weighted aggregation—denoted by —of all the individual random variables ( ), then