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Example 6

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


based on sample value calculations (2.4:

(2.10)

derived from (2.2) and (2.3):


If and are to be treated as functions of a continuous variable for , this suggests calculations or estimates on the lines of

(2.11)

where is a partition of .

For Example 5 the sample calculation (2.4) of total portfolio value leads unproblematically to the random variable representation (2.5), . Though we have not yet settled on a meaning for stochastic integral, the discrete expression


points towards 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)

where for .

But (2.11) has , not the 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 . Why is this?

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


The latter corresponds to the calculation


of Example 5, where is used, but not or any value intermediate between and .

The reasoning is as follows. At time the investor makes a policy decision to purchase a quantity of shares whose value from time up to (but not including) time is . This number of shares (the portfolio) is retained up to time . At that instant of time the decision cycle is repeated, and the investor adjusts the portfolio by taking a position of holding number of shares, each of which has the new value .

In the time period to , the gain in value of the portfolio level chosen at time is


not , since the portfolio quantity operates in the time period to (not to ). Reverting to continuous form, this translates to Riemann sum terms of the form


Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics

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