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(2.17)
As in previous versions of the sample space
for measurable sets
2.5 More on Basic Stochastic Integral
The constructions in Sections 2.3 and 2.4 purported to be about stochastic integration. While a case can be made that (2.6) and (2.7) are actually stochastic integrals, such simple examples are not really what the standard or classical theory of Chapter 1 is all about. The examples and illustrations in Sections 2.3 and 2.4 may not really be much help in coming to grips with the standard theory of stochastic integrals outlined in Chapter 1.
This is because Chapter 1, on the definition and meaning of classical stochastic integration, involves subtle passages to a limit, whereas (2.6) and (2.7) involve only finite sums and some elementary probability calculations.
From the latter point of view, introducing probability measure spaces and random‐variables‐as‐measurable‐functions seems to be an unnecessary complication. So, from such a straightforward starting point, why does the theory become so challenging and “messy”, as portrayed in Chapter 1
As in Example 2, the illustration in Section 2.3 involves dividing up the time period (4 days) into 4 sections; leading to sample space
Other simplifications can be similarly adopted. For instance, only two kinds of changes are contemplated in Section 2.3: increase (Up) or decrease (Down). But that is merely a slight technical limitation. Just as the number of discrete times can be increased indefinitely, so can the number of distinct, discrete values which can be potentially taken at any instant.
There is a plausible argument for this essentially discrete approach, at least in the case of financial shareholding. Actual stock market values register changes at discrete intervals of time (time ticks), and the amount of change that can occur is measured in discrete divisions (or basis points) of the currency.
So why does the mathematical model for such processes (as described in Chapter 1, for instance) require passages to a limit involving infinite divisibility of both the time domain, and the value range?
In fact there are sound mathematical reasons for this seemingly complicated approach. For one thing, instead of choosing one of many possible finite division points of time and values, passage to a limit—if that is possible—replaces a multiplicity of rather arbitrary choices by a single definite procedure, which may actually be easier to compute.
Furthermore, Brownian motion provides a good mathematical model for many random processes, and Brownian motion is based on continuous time and continuous values, not discrete.
The
