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

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of integral Subscript 0 Superscript t Baseline.

integral Subscript 0 Superscript t Baseline script upper Z left-parenthesis s right-parenthesis d script upper X left-parenthesis s right-parenthesis equals sigma-summation Underscript j equals 1 Overscript n Endscripts left-parenthesis script upper Z Subscript j minus 1 Baseline integral Subscript t Subscript j minus 1 Baseline Superscript t Subscript j Baseline Baseline d script upper X left-parenthesis s right-parenthesis right-parenthesis equals sigma-summation Underscript j equals 1 Overscript n Endscripts left-parenthesis alpha Subscript j minus 1 Baseline integral Subscript t Subscript j minus 1 Baseline Superscript t Subscript j Baseline Baseline d script upper X left-parenthesis s right-parenthesis right-parenthesis question-mark

      With each alpha Subscript j Baseline equals 1, this would imply

      (1.1)integral Subscript 0 Superscript t Baseline d script upper X left-parenthesis s right-parenthesis equals script upper X left-parenthesis t right-parenthesis minus script upper X left-parenthesis 0 right-parenthesis equals script upper X left-parenthesis t right-parenthesis period

      If this is unproblematical, it should be possible to deduce it from one or other of the various mathematical definitions of Brownian motion left-parenthesis script upper X left-parenthesis s right-parenthesis right-parenthesis, along with some mathematical definition of the integral integral Subscript 0 Superscript t Baseline in this context.

      But it appears that there is no such understanding of integral Subscript 0 Superscript t Baseline d script upper X left-parenthesis s right-parenthesis equals script upper X left-parenthesis t right-parenthesis. 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, I2 has the following condition on the expected value of the integral of the process left-parenthesis script upper Z left-parenthesis s right-parenthesis squared:

normal upper E left-parenthesis integral Subscript 0 Superscript t Baseline left-parenthesis script upper Z left-parenthesis s right-parenthesis right-parenthesis squared d s right-parenthesis less-than infinity period

      The idea here is that, if script upper Y is the random entity obtained by carrying out some form of weighted aggregation—denoted by integral Subscript 0 Superscript t Baseline left-parenthesis script upper Z left-parenthesis s right-parenthesis right-parenthesis squared d s—of all the individual random variables script upper Z left-parenthesis s right-parenthesis (0 less-than-or-equal-to s less-than-or-equal-to t), then

normal upper E left-parenthesis script upper Y right-parenthesis equals integral Underscript normal upper Omega Endscripts script upper Y left-parenthesis omega right-parenthesis d upper P less-than infinity period

      This formulation assumes that the aggregative operation integral Subscript 0 Superscript t Baseline left-parenthesis script upper Z left-parenthesis s right-parenthesis right-parenthesis squared d s, involving infinitely many random variables script upper Z left-parenthesis s right-parenthesis (0 less-than-or-equal-to s less-than-or-equal-to t), produces a single random entity script upper Y whose expected value can be obtained by means of the operation integral Underscript normal upper Omega Endscripts midline-horizontal-ellipsis d upper P.

      Additionally, integral Subscript 0 Superscript t Baseline script upper Z left-parenthesis s right-parenthesis squared d s is said to be a Lebesgue integral‐type construction. The integral Subscript 0 Superscript t Baseline midline-horizontal-ellipsis d s part of this statement should be unproblematical. The domain left-bracket 0 comma t right-bracket is a real interval, and has a distance or length function, which, in the context of Lebesgue integration on the domain, gives rise to Lebesgue measure mu on the space script upper M of Lebesgue measurable subsets of left-bracket 0 comma t right-bracket. So integral Subscript 0 Superscript t Baseline midline-horizontal-ellipsis d s can also be expressed as integral Subscript 0 Superscript t Baseline midline-horizontal-ellipsis d mu.

      However, the random variable‐valued integrand script upper Z left-parenthesis s right-parenthesis squared is less familiar in Lebesgue integration. Suppose, instead, that the integrand is a real‐number‐valued function f left-parenthesis s right-parenthesis. Then the Lebesgue integral integral Subscript 0 Superscript t Baseline f left-parenthesis s right-parenthesis d s, or integral Subscript 0 Superscript t Baseline f left-parenthesis s right-parenthesis d mu, is defined if the integrand function f is Lebesgue measurable. So if J is an interval of real numbers in the range of f, the set f Superscript negative 1 Baseline left-parenthesis upper J right-parenthesis is a member of the class script upper M of measurable sets; giving

f Superscript negative 1 Baseline left-parenthesis upper J right-parenthesis comma equals StartSet s colon f left-parenthesis <p style= Скачать книгу