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

, are the same function if and only if

      Does the definition of stochastic integral in I1, I2, I3 yield such a unique value for ? I2 and I3 do not guarantee uniqueness: there may be different sequences in I2 which converge “in mean square” to . In effect, I4 asserts weak convergence of the integrals of the step functions to a value for the integral of , that value being not necessarily unique.

      If the integral does not have a unique value, what connections may exist between alternative values? Suppose there is more than one candidate random variable, say and , for the value of the stochastic integral,

      In that case, what is the relation between and ? For instance, is it the case that, for each real number a, the probabilities of corresponding measurable sets are equal (such as ):

      The framework outlined above does not include the important case , where () is Brownian motion. Broadly speaking, means that the random variables represented by finite sums

      converge as tend to zero, each j. In fact the convergence is weak, not point‐wise, with

      and the weak 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 ,