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

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limit t is a fixed real number which can be regarded as a degenerate random variable. This result is basic to the construction I1, I2, I3, I4.

      A closer reading of source material may provide answers and/or corrections to some or all of the above comments and queries. Any misinterpretation, confusion, and errors may be dispelled by closer examination of the underlying ideas.

      Aside from these issues, and looking beyond the classical mathematical theory, the general idea of stochastic integral is, in intuitive terms, a persuasive, natural and practical way of thinking about the underlying reality.

      An alternative (and hopefully more understandable) mathematical way of representing this reality is presented in subsequent chapters of this book.

      Example 2

In order to focus on the underlying ideas, here is a simple illustration. Suppose, at different times t, (or ) is a random variable, with sample space . For simplicity suppose, for each t, has the same sample space and the same probability distribution. Suppose further that has only a finite number m of possible values , each equally likely. Then we can take

      and probability . For ,

      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

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