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assumed, such that, for all random variables and processes, the probability that any random variable has an outcome in a particular set
can be calculated using the appropriate technical calculation
1 relevant to each random variable. If the random variables or processes have a time structure, then mathematical properties of
filtration and
adaptedness ensure that sets
A which qualify as
‐measurable events at earlier times will still qualify as such at subsequent times.
The integrator is a random variable. The integrand function or is also a random variable. And the (stochastic) integral , is a random variable. This point is sometimes illustrated in textbooks by means of examples such as the following.
Example 1
Suppose (a random quantity) is the price of an asset at time t. Then, for times , is the change in the price of the asset, the change or difference also being random. Suppose the quantity of asset holding (sometimes denoted as ) is unpredictable or random. The product of these two,
is then a random variable representing the change in the value of the total asset holding. The stochastic integral , represents the aggregate or sum of these changes over the period of time ; and is a random variable.
Here, use of the symbol for the integrand (instead of the usual ) indicates that while the integrand is a random variable dependent on s, it does not necessarily depend on the integrator random variable . If, in fact, there is such dependence, then an appropriate notation2 for the integrand is .
The notation and terminology of ordinary integration is used in I1, I2, I3, I4, and they provide a certain “feel” for what is going on. But the various elements of the system are clearly different from ordinary integration. Can we get some more precise idea of what is really going on?
The “integration‐like” construction in I1 suggests that the domain of integration is , and that the integrand takes values in a class of functions (—random variables; that is, functions which are measurable with respect to some probability space, or spaces).
How does this compare with more familiar integration scenarios? Basic integration (“”) has two elements: firstly, a domain of integration containing values of the integration variable s, and secondly, an integrand function which depends on the values s in the domain of integration. The more familiar integrand functions have values which are real or complex numbers ; and which are deterministic (that is, “definite”, not approximate or estimated).
The construction in I1, I2, I3 indicates an integration domain or . (There is nothing surprising in that.) But in I1, I2, I3 the integrand values are not real or complex numbers, but random variables—which may be a bit surprising.
But it is not unprecedented. For instance, the Bochner integration process in mathematical analysis deals with integrands whose values are functions, not numbers.
The construction and definition of the Bochner integral [105] is similar in some respects to the classical Itô integral. What is the end result of the construction in I1, I2, I3?
In general, the integral of a function f gives a kind of average or aggregation of all the possible values of f. So if each value of the integrand is a random variable, the integral of f should itself be a random variable—that is, a function which is measurable with respect to an underlying probability measure space.
If the notation is valid or justifiable for the stochastic integral, it suggests that the Itô integral construction derives a single random variable
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