Patrick Muldowney

Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics

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alt="upper P left-parenthesis upper Y Subscript t Baseline element-of upper A right-parenthesis equals StartFraction StartAbsoluteValue upper A EndAbsoluteValue Over m EndFraction comma"/>

      where StartAbsoluteValue upper A EndAbsoluteValue is the number of elements in A. Then, for each t, upper Y Subscript t is a left-parenthesis normal upper Omega comma upper P right-parenthesis‐measurable function and thus a random variable. (We may also suppose, if it is convenient for us, that for any t, t prime, the random variables upper Y Subscript t Baseline comma upper Y Subscript t prime Baseline are independent.)

      Now suppose that, for 0 less-than t less-than-or-equal-to tau, upper Z Subscript t is another indeterminate or unpredictable quantity; and that, for given t, the possible values of upper Z Subscript t depend in some deterministic way on the corresponding values of upper Y Subscript t, so

upper Z Subscript t Baseline equals f left-parenthesis upper Y Subscript t Baseline right-parenthesis

      where f is a deterministic function. For instance, the deterministic relation could be upper Z Subscript t Baseline equals upper Y Subscript t Superscript 2, so if the value taken by upper Y Subscript t at time t is y Subscript t, then the value that upper Z Subscript t takes is y Subscript t Superscript 2. Provided f is a “reasonably nice” function (such as left-parenthesis dot right-parenthesis squared), then upper Z Subscript t is measurable with respect to left-parenthesis normal upper Omega comma upper P right-parenthesis, 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 upper Z Subscript t with respect to upper Y Subscript t. Essentially, with normal upper Omega equals StartSet 1 comma ellipsis comma m EndSet, then for each t, for omega equals i element-of normal upper Omega, and for i equals 1 comma ellipsis comma m,

upper Z Subscript t Baseline left-parenthesis omega right-parenthesis equals f left-parenthesis upper Y Subscript t Baseline left-parenthesis omega right-parenthesis right-parenthesis comma or upper Z Subscript t Baseline left-parenthesis i right-parenthesis equals f left-parenthesis upper Y Subscript t Baseline left-parenthesis i right-parenthesis right-parenthesis comma

      the two formulations being equivalent. If the stochastic integral “integral Subscript 0 Superscript tau Baseline upper Z Subscript t Baseline d upper Y Subscript t” is to be formulated in terms of Lebesgue integrals in 0 less-than t less-than-or-equal-to tau (as intimated in I1, I2, I3, I4), then some properties of t‐measurability (0 less-than-or-equal-to t less-than-or-equal-to tau) are suggested. This aspect can also be simplified, as follows.

      Just as normal upper Omega was reduced to a finite number m of possible values, left-bracket 0 comma tau right-bracket can be replaced by a finite number of fixed time values 0 less-than tau 1 less-than tau 2 less-than midline-horizontal-ellipsis less-than tau Subscript n Baseline equals tau if the family of random variables upper Y Subscript t (0 less-than t less-than-or-equal-to tau) is replaced by upper Y Subscript tau Sub Subscript j (1 less-than-or-equal-to j less-than-or-equal-to n); so there are only a finite number n of random variables upper Y Subscript j Baseline equals upper Y Subscript tau Sub Subscript j,

upper Y Subscript t Baseline equals upper Y Subscript tau Sub Subscript j Subscript Baseline equals upper Y Subscript j Baseline for tau Subscript j minus 1 Baseline less-than t less-than-or-equal-to tau Subscript j Baseline comma 1 less-than-or-equal-to j less-than-or-equal-to n semicolon

      and the random variables can be written

upper Z Subscript t Baseline left-parenthesis omega right-parenthesis equals upper Z Subscript t Baseline left-parenthesis i right-parenthesis equals f left-parenthesis upper Y Subscript t Baseline left-parenthesis omega right-parenthesis right-parenthesis equals f left-parenthesis upper Y Subscript t Baseline left-parenthesis i right-parenthesis right-parenthesis equals f left-parenthesis upper Y Subscript j Baseline left-parenthesis i right-parenthesis right-parenthesis