Product integral

A product integral is any product-based counterpart of the usual sum-based integral of calculus. The first product integral ("" below) was developed by the mathematician Vito Volterra in 1887 to solve systems of linear differential equations. Other examples of product integrals are the (' below), the (' below), and some other integrals of non-Newtonian calculus.
Product integrals have found use in areas from epidemiology (the Kaplan–Meier estimator) to stochastic population dynamics using multiplication integrals (multigrals), analysis and quantum mechanics. The , together with the geometric derivative, is useful in image analysis and in the study of growth/decay phenomena (e.g., in economic growth, bacterial growth, and radioactive decay). 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" formula_1 notation for product integration instead of the "integral" formula_2 (usually modified by a superimposed "times" symbol or letter P) 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 formula_3 can be defined by the relation
where the limit is taken over all partitions of the interval formula_5 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 formula_7:
which is not a multiplicative operator. (So the concepts of product integral and multiplicative integral are not the same).
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 (see the references below).
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 (i.e. normal products applied from left)
Right Product Integral
With the notation of right products (i.e. applied from right)
Where formula_13 is the identity matrix and D is a partition of the interval [a,b] 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
(with formula_16) and the multiplicative analog to the (normal/standard/additive) integral
(with formula_18):
It is very useful in stochastics, where the log-likelihood (i.e. the logarithm of a product integral of independent random variables) equals the integral of the logarithm of these (infinitesimally many) 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.
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 inverse of the geometric integral, which is the geometric derivative, denoted formula_26, is defined using the following relationship:
Thus, the following can be concluded:
where X is a random variable with probability distribution "F"("x").
Compare with the standard law of large numbers:
Lebesgue-type product-integrals.
Just like the Lebesgue version of (classical) 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 formula_33 with corresponding partition formula_34 and a tagged partition
one approximation of the "Riemann definition" of the is given by
The (type I) 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 formula_38 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 formula_38, 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 (type I) product integral" of a step function as
where formula_41 is a tagged partition, and again formula_34 is the partition corresponding to the step function formula_38. (In contrast, the corresponding quantity would not be unambiguously defined using the first type of approximation.)
This generalizes to arbitrary measure spaces readily. If formula_44 is a measure space with measure formula_45, then for any product-integrable simple function formula_46 (i.e. a conical combination of the indicator functions for some disjoint measurable sets formula_47), its type I product integral is defined to be
since formula_49 is the value of formula_38 at any point of formula_51. In the special case where formula_52, formula_53 is Lebesgue measure, and all of the measurable sets formula_51 are intervals, one can verify that this is equal to the definition given above for that special case. Analogous to the theory of Lebesgue (classical) integrals, the of any product-integrable function formula_38 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 formula_38:
where we used the definition of integral for simple functions. Moreover, because continuous functions like formula_59 can be interchanged with limits, and the product integral of any product-integrable function formula_38 is equal to the limit of product integrals of simple functions, it follows that the relationship
holds generally for "any" product-integrable formula_38. 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 formula_38 one can define a set function formula_64 by defining, for every measurable set formula_65,
where formula_67 denotes the indicator function of formula_68. Then for any two "disjoint" measurable sets formula_69 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 formula_71, and a measurable set formula_72, it is generally the case that
Type II: geometric integral.
If formula_44 is a measure space with measure formula_45, then for any product-integrable simple function formula_46 (i.e. a conical combination of the indicator functions for some disjoint measurable sets formula_47), 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 formula_38:
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 (classical) integrals. In other words, because continuous functions like formula_59 and formula_82 can be interchanged with limits, and the product integral of any product-integrable function formula_38 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 formula_38. This generalizes the property of mentioned above.