The product integral was developed by the mathematician Vito Volterra in 1887 to solve systems of linear differential equations.
The non-commutative case is far more complicated; it requires proper path-ordering to make the integral well-defined.
For the commutative case, three distinct definitions are commonplace in the literature, referred to as Type-I, Type-II or geometric, and type-III or bigeometric.
[3][4][5] Such integrals have found use in epidemiology (the Kaplan–Meier estimator) and stochastic population dynamics.
The integrand is generally an operator belonging to some non-commutative algebra.
Examples include the Dyson expansion, the integrals that occur in the operator product expansion and the Wilson line, a product integral over a gauge field.
The product integral also occurs in control theory, as the Peano–Baker series describing state transitions in linear systems written in a master equation type form.
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.
Note how in this case time ordering becomes evident in the definitions.
The product integral satisfies a collection of properties defining a one-parameter continuous group; these are stated in two articles showing applications: the Dyson series and the Peano–Baker series.
The commutative case is vastly simpler, and, as a result, a large variety of distinct notations and definitions have appeared.
(usually modified by a superimposed times symbol or letter P) favoured by Volterra and others.
An arbitrary classification of types is adopted to impose some order in the field.
The type I product integral corresponds to Volterra's original definition.
The logarithm is well-defined if f takes values in the real or complex numbers, or if f takes values in a commutative field of commuting trace-class operators.
, 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: When the integrand takes values in the real numbers, then the product intervals become easy to work with by using simple functions.
Another approximation of the "Riemann definition" of the type I product integral is defined as When
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[21] the "Lebesgue (type I) product integral" of a step function as where
(In contrast, the corresponding quantity would not be unambiguously defined using the first type of approximation.)
(i.e. a conical combination of the indicator functions for some disjoint measurable sets
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
is equal to the limit of product integrals of simple functions, it follows that the relationship holds generally for any product-integrable
The Type I integral is multiplicative as a set function,[22] which can be shown using the above property.
one has This property can be contrasted with measures, which are sigma-additive set functions.
(i.e. a conical combination of the indicator functions for some disjoint measurable sets
), its type II product integral 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
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