Читать книгу Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics - Patrick Muldowney - Страница 16

In order to focus on the underlying ideas, here is a simple illustration. Suppose, at different times t, (or ) is a random variable, with sample space . For simplicity suppose, for each t, has the same sample space and the same probability distribution. Suppose further that has only a finite number m of possible values , each equally likely. Then we can take

and probability . For ,

where is the number of elements in A. Then, for each t, is a ‐measurable function and thus a random variable. (We may also suppose, if it is convenient for us, that for any t, , the random variables are independent.)

Now suppose that, for , is another indeterminate or unpredictable quantity; and that, for given t, the possible values of depend in some deterministic way on the corresponding values of , so

where f is a deterministic function. For instance, the deterministic relation could be , so if the value taken by at time t is , then the value that takes is . Provided f is a “reasonably nice” function (such as ), then is measurable with respect to , and is itself a random variable.

This scenario is in broad conformity with I1, I2, I3, I4 above. So it may be possible to consider, in those terms, the stochastic integral of with respect to . Essentially, with , then for each t, for , and for ,

the two formulations being equivalent. If the stochastic integral “” is to be formulated in terms of Lebesgue integrals in (as intimated in I1, I2, I3, I4), then some properties of t‐measurability () are suggested. This aspect can also be simplified, as follows.

Just as was reduced to a finite number m of possible values, can be replaced by a finite number of fixed time values if the family of random variables () is replaced by (); so there are only a finite number n of random variables ,

and the random variables can be written

for , . (Below, will be taken to be .) Replacing the domains and by and , respectively, ensures measurability in and t. It also ensures measurability for the conditional cases of (or ) with already determined as known real numbers when .

Let represent sample values (or potential occurrences) of the random variables . For any given j, i can have value

so is the set of permutations, with repetition, of the numbers taken n at a time.

The ideas in I1, I2, I3, I4) suggest the following sample values for the stochastic integral (or “”):

The subscript labels the random variability in this calculation, and demonstrates that this version of the stochastic integral can take possible values; though not all of the possible values are necessarily distinct.

For further simplification, take and ; so, at each of times (), the random variable can take one of two possible values, . Then, by enumerating the permutations with repetition of things taken at a time , the 8 possible sample values of the stochastic integral are:

Now suppose that the deterministic function f is exponentiation to the power of 2 (so ); and suppose the random variable (or above) has sample values and with equal probabilities . Calculating each of the above expressions, the 8 sample evaluations of the stochastic integral are, respectively,

each having equal probability; so the distinct sample values of the random variable X are , with equal probabilities . Thus it happens, in this case, that the stochastic integral has the same sample space , and the same probabilities, as each of the random variables .

This example gets round the technical problem of measurability by discretizing the domains and , and using only function , which are step functions in respect of their dependence on t and .

This discrete or step function device is a fairly standard ploy of mathematical analysis. Following through on this device usually involves moving on to functions which are limits of step functions; and this procedure generally involves use of some conditions which ensure that the integral of a “limit of step functions” is equal to the limit of the integrals of the step functions.

Broadly speaking, it is not unreasonable to anticipate that this approach will succeed for measurable (or “reasonably nice”) functions—such as functions which are “smooth”, or which are continuous.

But the full meaning of measurability is quite technical, involving infinite operations on sigma‐algebras of sets. This can make the analysis difficult.

Accordingly, it may be beneficial to seek an alternative approach to the analysis of random variation for which measurability is not the primary or fundamental starting point. Such an alternative is demonstrated in Chapters 2 and 3, leading to an alternative exposition of stochastic integration in ensuing chapters.

The method of exposition is slow and gradual, starting with the simplest models and examples. The step‐by‐step approach is as follows.

 Though there are other forms of stochastic integral, the focus will be on where Z and X are stochastic processes.

 The sample space will generally be where is the set of real numbers, is an indexing set such as the real interval , and is a cartesian product.

 Z and X are stochastic process , (); and g is a deterministic function. More often, the process Z is X, so the stochastic integral⁶ is .

 The approach followed in the exposition is to build up to such stochastic integrals by means of simpler preliminary examples, broadly on the following lines:– Initially take to be a finite set, then a countable set, then an uncountable set such as .– Initially, let the process(es) X (and/or Z) be very easy versions of random variation, with only a finite number of possible sample values.– Similarly let the integrand g be an easily calculated function, such as a constant function or a step function.– Gradually increase the level of “sophistication”, up to the level of recognizable stochastic integrals.

 This progression helps to develop a more robust intuition for this area of random variation. On that basis, the concept of “stochastic sums” is introduced. These are more flexible and more far reaching than stochastic integrals; and, unlike the latter, they are not over‐burdened with issues involving weak convergence.