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

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which follows naturally from the naive or realistic approach described above, and which does not require the theory of measure as its foundation. The following pages are intended to convey the basic ideas of this approach.

      Example 6

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

      based on sample value calculations (2.4:

      (2.10)w left-parenthesis t right-parenthesis equals sigma-summation Underscript s equals 1 Overscript t Endscripts z left-parenthesis s minus 1 right-parenthesis left-parenthesis x left-parenthesis s right-parenthesis minus x left-parenthesis s minus 1 right-parenthesis right-parenthesis comma

c left-parenthesis s right-parenthesis equals z left-parenthesis s minus 1 right-parenthesis times left-parenthesis x left-parenthesis s right-parenthesis minus x left-parenthesis s minus 1 right-parenthesis right-parenthesis comma w left-parenthesis s right-parenthesis equals w left-parenthesis s minus 1 right-parenthesis plus c left-parenthesis s right-parenthesis period

      If x left-parenthesis s right-parenthesis and z left-parenthesis s right-parenthesis are to be treated as functions of a continuous variable s for 0 less-than-or-equal-to s less-than-or-equal-to t, this suggests calculations or estimates on the lines of

      where 0 equals s 0 less-than s 1 less-than midline-horizontal-ellipsis less-than s Subscript n Baseline equals t is a partition of left-bracket 0 comma t right-bracket.

      For Example 5 the sample calculation (2.4) of total portfolio value leads unproblematically to the random variable representation (2.5), upper W left-parenthesis t right-parenthesis equals integral Subscript 0 Superscript t Baseline upper Z left-parenthesis s right-parenthesis d upper X left-parenthesis s right-parenthesis. Though we have not yet settled on a meaning for stochastic integral, the discrete expression

upper W left-parenthesis t right-parenthesis equals sigma-summation Underscript j equals 1 Overscript n Endscripts upper Z left-parenthesis s Subscript j minus 1 Baseline right-parenthesis left-parenthesis upper X left-parenthesis s Subscript j Baseline right-parenthesis minus upper X left-parenthesis s Subscript j minus 1 Baseline right-parenthesis right-parenthesis comma

      points towards integral Subscript 0 Superscript t Baseline z left-parenthesis s right-parenthesis d x left-parenthesis s right-parenthesis as a continuous variable form of stochastic integral. It seems that the sample value form of the latter should be the Riemann‐Stieltjes integral integral Subscript 0 Superscript t Baseline z left-parenthesis s right-parenthesis d x left-parenthesis s right-parenthesis, for which a Riemann sum estimate is

      where s Subscript j minus 1 Baseline less-than-or-equal-to s prime Subscript j Baseline less-than-or-equal-to s Subscript j for 1 less-than-or-equal-to j less-than-or-equal-to n.

      The issue is to choose between two forms of Riemann sum:

w left-parenthesis t right-parenthesis equals sigma-summation Underscript j equals 1 Overscript n Endscripts z left-parenthesis s prime Subscript j right-parenthesis left-parenthesis x left-parenthesis s Subscript j Baseline right-parenthesis minus x left-parenthesis s Subscript j minus 1 Baseline right-parenthesis right-parenthesis comma w left-parenthesis t right-parenthesis equals sigma-summation Underscript j equals 1 Overscript n Endscripts z left-parenthesis s Subscript j minus 1 Baseline right-parenthesis left-parenthesis x left-parenthesis s Subscript j Baseline <p style= Скачать книгу