Product integral


A product integral is any product-based counterpart of the usual sum-based integral of calculus. The first product integral was developed by the mathematician Vito Volterra in 1887 to solve systems of linear differential equations. Other examples of product integrals are the , the , and some other integrals of non-Newtonian calculus.
Product integrals have found use in areas from epidemiology to stochastic population dynamics using multiplication integrals, analysis and quantum mechanics. The, together with the geometric derivative, is useful in image analysis and in the study of growth/decay phenomena. The, together with the bigeometric derivative, is useful in some applications of fractals, and in the theory of elasticity in economics.
This article adopts the "product" notation for product integration instead of the "integral" favoured by Volterra and others. An arbitrary classification of types is also adopted to impose some order in the field.

Basic definitions

The classical Riemann integral of a function can be defined by the relation
where the limit is taken over all partitions of the interval whose norms approach zero.
Roughly speaking, product integrals are similar, but take the limit of a product instead of the limit of a sum. They can be thought of as "continuous" versions of "discrete" products.
The most popular product integrals are the following:

Type I: Volterra integral

The type I product integral corresponds to Volterra's original definition. The following relationship exists for scalar functions :
which is not a multiplicative operator..
The Volterra product integral is most useful when applied to matrix-valued functions or functions with values in a Banach algebra, where the last equality is no longer true.
When applied to scalars belonging to a non-commutative field, to matrixes, and to operators, i.e. to mathematical objects that don't commute, the Volterra integral splits in two definitions
Left Product integral
With the notation of left products
Right Product Integral
With the notation of right products
Where is the identity matrix and D is a partition of the interval in the Riemann sense, i.e. the limit is over the maximum interval in the partition.
Note how in this case time ordering comes evident in the definitions.
For scalar functions, the derivative in the Volterra system is the logarithmic derivative, and so the Volterra system is not a multiplicative calculus and is not a non-Newtonian calculus.

Type II: geometric integral

which is called the geometric integral and is a multiplicative operator.
This definition of the product integral is the continuous analog of the discrete product operator
and the multiplicative analog to the integral
:
It is very useful in stochastics, where the log-likelihood equals the integral of the logarithm of these random variables:

Type III: bigeometric integral

where r = ln a, and s = ln b.
The type III product integral is called the bigeometric integral and is a multiplicative operator.

Results

;Basic results
The following results are for the. Other types produce other results.
The plays a central role in the geometric calculus, which is a multiplicative calculus.
;The fundamental theorem
where is the geometric derivative.
;Product rule
;Quotient rule
;Law of large numbers
where X is a random variable with probability distribution F.
Compare with the standard law of large numbers:

Lebesgue-type product-integrals

Just like the Lebesgue version of integrals, one can compute product integrals by approximating them with the product integrals of simple functions. Each type of product integral has a different form for simple functions.

Type I: Volterra integral

Because simple functions generalize step functions, in what follows we will only consider the special case of simple functions that are step functions. This will also make it easier to compare the Lebesgue definition with the Riemann definition.
Given a step function with corresponding partition and a tagged partition
one approximation of the "Riemann definition" of the is given by
The product integral was defined to be, roughly speaking, the limit of these products by Ludwig Schlesinger in a 1931 article.
Another approximation of the "Riemann definition" of the type I product integral is defined as
When is a constant function, the limit of the first type of approximation is equal to the second type of approximation. Notice that in general, for a step function, the value of the second type of approximation doesn't depend on the partition, as long as the partition is a refinement of the partition defining the step function, whereas the value of the first type of approximation does depend on the fineness of the partition, even when it is a refinement of the partition defining the step function.
It turns out that that for any product-integrable function, the limit of the first type of approximation equals the limit of the second type of approximation. Since, for step functions, the value of the second type of approximation doesn't depend on the fineness of the partition for partitions "fine enough", it makes sense to define the "Lebesgue product integral" of a step function as
where is a tagged partition, and again is the partition corresponding to the step function.
This generalizes to arbitrary measure spaces readily. If is a measure space with measure, then for any product-integrable simple function , its type I product integral is defined to be
since is the value of at any point of. In the special case where, is Lebesgue measure, and all of the measurable sets are intervals, one can verify that this is equal to the definition given above for that special case. Analogous to the theory of Lebesgue integrals, the of any product-integrable function can be written as the limit of an increasing sequence of Volterra product integrals of product-integrable simple functions.
Taking logarithms of both sides of the above definition, one gets that for any product-integrable simple function :
where we used the definition of integral for simple functions. Moreover, because continuous functions like can be interchanged with limits, and the product integral of any product-integrable function is equal to the limit of product integrals of simple functions, it follows that the relationship
holds generally for any product-integrable. This clearly generalizes the property.
The is multiplicative as a set function, which can be shown using the above property. More specifically, given a product-integrable function one can define a set function by defining, for every measurable set,
where denotes the indicator function of. Then for any two disjoint measurable sets one has
This property can be contrasted with measures, which are additive set functions.
However the is not multiplicative as a functional. Given two product-integrable functions, and a measurable set, it is generally the case that

Type II: geometric integral

If is a measure space with measure, then for any product-integrable simple function , its is defined to be
This can be seen to generalize the definition given above.
Taking logarithms of both sides, we see that for any product-integrable simple function :
where we have used the definition of the Lebesgue integral for simple functions. This observation, analogous to the one already made, allows one to entirely reduce the "Lebesgue theory of " to the Lebesgue theory of integrals. In other words, because continuous functions like and can be interchanged with limits, and the product integral of any product-integrable function is equal to the limit of some increasing sequence of product integrals of simple functions, it follows that the relationship
holds generally for any product-integrable. This generalizes the property of mentioned above.