Читать книгу Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics - Patrick Muldowney - Страница 15

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 (or ) from many jointly varying random variables, such as , as varies between the values 0 and t. This is reminiscent of Norbert Wiener's construction in [169], which is in some sense a mathematical replication in one dimension of Brownian motion; even though the latter is essentially an infinite‐dimensional phenomenon with infinitely many variables. Without losing any essential information, a situation involving infinitely many variables is converted to a scenario involving only one variable.³

The proof of the Itô isometry relation (see I1) indicates that, as a stochastic process, must be independent of . Otherwise the construction I1, I2, I3 would seem to be inadequate as it stands, whenever the process is replaced by a process .

In I3 the integrand does not have step function form; and, on the face of it, indicates dependence of (or ) on random variables and for every s, . If the integrand were (which, in general, it is not), with joint random variability for , and if is Brownian motion, then the joint probability space for the processes and is given by the Wiener probability measure and its associated multi‐dimensional measure space. (The latter are described in Chapter 5 below.)

Returning to I1, the Itô integral of step function is defined as

where the are random variable values of . It is perfectly valid to combine finite numbers of random variables in this way, in order to produce, as outcome, a single random variable (—which may be a joint random variable depending on many underlying random variables).

This part of the formulation of the integral of a step function in I1 corresponds to the integral of a step function in basic integration, and does not require any passage to a limit of random variables.

Now suppose each is a fixed real number ; so, for , . (Accordingly, in I1, can be regarded as a “degenerate” random variable, with atomic probability value.) Suppose the integrator is the real‐valued ds instead of the random variable‐valued . Then⁴

Formally, at least, this looks like the definition in I1 of when is a step function. The factor equals for each j. This emerges naturally from the mathematical meaning of the length or distance variable s, and from the mathematical meaning of .

Can this be replicated in I1 when is a step function, or when each is a fixed real number ? Is it the case that

With each , this would imply

(1.1)

If this is unproblematical, it should be possible to deduce it from one or other of the various mathematical definitions of Brownian motion , along with some mathematical definition of the integral in this context.

But it appears that there is no such understanding of . So, as in I1, it seems that this formulation is to be regarded as a basic postulate or axiom of stochastic integration.

Returning to the definition of the classical Itô integral, I2 has the following condition on the expected value of the integral of the process :

The idea here is that, if is the random entity obtained by carrying out some form of weighted aggregation—denoted by —of all the individual random variables (), then

This formulation assumes that the aggregative operation , involving infinitely many random variables (), produces a single random entity whose expected value can be obtained by means of the operation .

Additionally, is said to be a Lebesgue integral‐type construction. The part of this statement should be unproblematical. The domain is a real interval, and has a distance or length function, which, in the context of Lebesgue integration on the domain, gives rise to Lebesgue measure on the space of Lebesgue measurable subsets of . So can also be expressed as .

However, the random variable‐valued integrand is less familiar in Lebesgue integration. Suppose, instead, that the integrand is a real‐number‐valued function . Then the Lebesgue integral , or , is defined if the integrand function f is Lebesgue measurable. So if J is an interval of real numbers in the range of f, the set is a member of the class of measurable sets; giving

That is, for each J, is a Lebesgue measurable subset of . This is valid if, for instance, f is a continuous function of s, or if f is the limit of a sequence of step functions.

How does this translate to a random variable‐valued integrand such as ? Two kinds of measurability arise here, because, in addition to being a ‐measurable function of , is a random variable (as is ), and is therefore a P‐measurable function on the sample space :

Likewise . For to be meaningful as a Lebesgue‐type integral, the integrand must be ‐measurable (or ‐measurable) in some sense. At least, for purpose of measurability there needs to be some metric in the space of ‐measurable functions , , with , :

For example, the “distance” between and could be

With such a metric at hand, it may then be possible to define , or , as the limit of the integrals of (integrable) step functions converging to for , as .

Unfortunately, most standard textbooks do not give this point much attention. But for relatively straightforward integrands such as , it should not be too difficult.

Continuing the discussion of I1, I2, I3, I4, it appears that the output of this definition of stochastic integral is a random entity ; perhaps a process which is some collection of random variables .

Again comparing this with basic integration of a real number‐valued function , the integral is some kind of average or weighted aggregate value for . This integral, if it exists, produces a single unique real number (depending on the value of t), denoted by .

For random variable‐valued integrand , suppose (for the purpose of speculation) that the stochastic integral

(if it exists) is equivalent (in some unspecified sense) to a single, unique random variable . Remember, a random variable is a function, usually real‐valued⁵, defined on a sample space . Two such functions, and , are the same function if and only if

Does the definition of stochastic integral in I1, I2, I3 yield such a unique value for ? I2 and I3 do not guarantee uniqueness: there may be different sequences in I2 which converge “in mean square” to . In effect, I4 asserts weak convergence of the integrals of the step functions to a value for the integral of , that value being not necessarily unique.

If the integral does not have a unique value, what connections may exist between alternative values? Suppose there is more than one candidate random variable, say and , for the value of the stochastic integral,

In that case, what is the relation between and ? For instance, is it the case that, for each real number a, the probabilities of corresponding measurable sets are equal (such as ):

The framework outlined above does not include the important case , where () is Brownian motion. Broadly speaking, means that the random variables represented by finite sums

converge as tend to zero, each j. In fact the convergence is weak, not point‐wise, with

and the weak limit t is a fixed real number which can be regarded as a degenerate random variable. This result is basic to the construction I1, I2, I3, I4.

A closer reading of source material may provide answers and/or corrections to some or all of the above comments and queries. Any misinterpretation, confusion, and errors may be dispelled by closer examination of the underlying ideas.

Aside from these issues, and looking beyond the classical mathematical theory, the general idea of stochastic integral is, in intuitive terms, a persuasive, natural and practical way of thinking about the underlying reality.

An alternative (and hopefully more understandable) mathematical way of representing this reality is presented in subsequent chapters of this book.