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

Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics

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.

Before moving on to this, here is an elaboration of a technical point of a financial character, which appeared in Example 5 above and in the ensuing discussion, and which is relevant in stochastic integration.

Example 6

Expression (2.5) above gives two representations of a stochastic integral,

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

derived from (2.2) and (2.3):

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 for 0 less-than-or-equal-to s less-than-or-equal-to t , this suggests calculations or estimates on the lines of

(2.11) 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 right-parenthesis minus x left-parenthesis s Subscript j minus 1 Baseline right-parenthesis right-parenthesis comma

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

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 , for which a Riemann sum estimate is

(2.12) 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

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 .

But (2.11) has z left-parenthesis s Subscript j minus 1 Baseline right-parenthesis , not the z left-parenthesis s prime Subscript j right-parenthesis of (2.12). The logic of Example 5 indicates that only the left hand value is permitted in the Riemann sum estimates of the stochastic integral integral Subscript 0 Superscript t Baseline z left-parenthesis s right-parenthesis d x left-parenthesis s right-parenthesis . Why is this?

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 <hr><noindex><a href=

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