such , define its stochastic integral with respect to the process as
I4 If is Brownian motion the latter limit exists.
An objective of this book is to provide an alternative to the classical theory, not develop it. Thus the commentary, interpretation, and speculation of this section can be safely omitted by anybody who is either already familiar with, or is not interested in, the standard theory of stochastic integration.
Regarding notation, many textbooks use the symbol B for Brownian motion, whereas is used above. Textbooks also use the symbol for the integrand, where is used above. The reason for using notation instead of ) is to emphasise that the value of the integrand function is generally a random variable depending on s, and not generally a single, definite real or complex number (such as the deterministic function , for instance) of the kind which occurs in ordinary integration.
In classical probability theory, an underlying mathematical probability measure space is 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 calculation1 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
such , define its stochastic integral with respect to the process as
