For transition sample path number 7, UDDU, the overall gain in shareholding value is
where is a “negative gain” or net loss. With , this can be interpreted as the Stieltjes integral3
Observe that the number of shares held at any time depends on whether the share price has moved up or down. So , , is a deterministic function of ; and the value of varies randomly because varies randomly.
The same applies to the values of , including the terminal value , or with . Table 2.5 gives the respective process sample paths for processes, where the underlying share price process follows sequence DUUU (sample number 2 in Table 2.4).
Regarding notation, the symbols , (and so on) are used here, in contrast to symbols etc. which were used in discussion of stochastic calculus in Chapter 1 . In the latter, the emphasis was on the classical rigorous theory in which random variables are measurable functions, and this is signalled by using instead of , etc.
Where (rather than etc.) is used, the purpose is to indicate the “naive” or “natural” outlook which sees random variability, not in terms of abstract mathematical measurable sets and functions, but in terms of actual occurrences such as tossing a coin, or such as
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