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

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upper J EndSet comma element-of script upper M period"/>

      How does this translate to a random variable‐valued integrand such as script upper Z left-parenthesis s right-parenthesis squared? Two kinds of measurability arise here, because, in addition to being a mu‐measurable function of s element-of left-bracket 0 comma t right-bracket, script upper Z left-parenthesis s right-parenthesis is a random variable (as is script upper Z left-parenthesis s right-parenthesis squared), and is therefore a P‐measurable function on the sample space normal upper Omega:

StartLayout 1st Row 1st Column left-bracket 0 comma t right-bracket times normal upper Omega 2nd Column right-arrow from bar Overscript script upper Z Endscripts 3rd Column bold upper R comma 2nd Row 1st Column left-parenthesis s comma omega right-parenthesis 2nd Column right-arrow 3rd Column script upper Z left-parenthesis s comma omega right-parenthesis element-of bold upper R period EndLayout

      Likewise script upper Z left-parenthesis s right-parenthesis squared. For integral Subscript 0 Superscript 1 Baseline script upper Z left-parenthesis s right-parenthesis squared d s to be meaningful as a Lebesgue‐type integral, the integrand script upper Z left-parenthesis s right-parenthesis squared must be script upper M‐measurable (or mu‐measurable) in some sense. At least, for purpose of measurability there needs to be some metric in the space of left-parenthesis normal upper Omega comma upper P right-parenthesis‐measurable functions f Subscript s, 0 less-than-or-equal-to s less-than-or-equal-to t, with f Subscript s Baseline left-parenthesis omega right-parenthesis element-of bold upper R, omega element-of normal upper Omega:

StartSet f Subscript s Baseline equals script upper Z left-parenthesis s right-parenthesis squared colon 0 less-than-or-equal-to s less-than-or-equal-to t EndSet period

      For example, the “distance” between f Subscript s 1 and f Subscript s 2 could be

integral Underscript normal upper Omega Endscripts StartAbsoluteValue f Subscript s 1 Baseline left-parenthesis omega right-parenthesis minus f Subscript s 2 Baseline left-parenthesis omega right-parenthesis EndAbsoluteValue d upper P period

      With such a metric at hand, it may then be possible to define integral Subscript 0 Superscript t Baseline f Subscript s Baseline d s, or integral Subscript 0 Superscript t Baseline script upper Z left-parenthesis s right-parenthesis squared d s, as the limit of the integrals of (integrable) step functions f Subscript s Superscript left-parenthesis p right-parenthesis converging to f Subscript s for 0 less-than-or-equal-to s less-than-or-equal-to t, as p right-arrow infinity.

      Unfortunately, most standard textbooks do not give this point much attention. But for relatively straightforward integrands such as script upper Z left-parenthesis s right-parenthesis squared, it should not be too difficult.

      Continuing the discussion of I1, I2, I3, I4, it appears that the output of this definition of stochastic integral is a random entity script upper Y; perhaps a process which is some collection of random variables left-parenthesis script upper Y left-parenthesis s right-parenthesis right-parenthesis.

      Again comparing this with basic integration of a real number‐valued function f left-parenthesis s right-parenthesis, the integral integral Subscript 0 Superscript t Baseline f left-parenthesis s right-parenthesis d s is some kind of average or weighted aggregate value for StartSet f left-parenthesis s right-parenthesis colon 0 less-than-or-equal-to s less-than-or-equal-to t EndSet. This integral, if it exists, produces a single unique real number (depending on the value of t), denoted by integral Subscript 0 Superscript t Baseline f left-parenthesis s right-parenthesis d s.

      For random variable‐valued integrand f left-parenthesis s right-parenthesis equals script upper Z left-parenthesis s right-parenthesis, suppose (for the purpose of speculation) that the stochastic integral

integral Subscript 0 Superscript t Baseline script upper Z left-parenthesis s right-parenthesis d script upper X left-parenthesis s right-parenthesis