Product integral

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A product integral is any product-based counterpart of the usual sum-based integral of calculus. The first product integral (Type I below) was developed by the mathematician Vito Volterra in 1887 to solve systems of linear differential equations.[1][2] Other examples of product integrals are the geometric integral (Type II below), the bigeometric integral (Type III below), and some other integrals of non-Newtonian calculus.[3][4][5]

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 geometric integral, together with the geometric derivative, is useful in image analysis[6] and in the study of growth/decay phenomena (e.g., in economic growth, bacterial growth, and radioactive decay).[7][8] The bigeometric integral, together with the bigeometric derivative, is useful in some applications of fractals,[9][10][11][12] and in the theory of elasticity in economics.[3][5][13]

This article adopts the "product" [math]\displaystyle{ \prod }[/math] notation for product integration instead of the "integral" [math]\displaystyle{ \int }[/math] (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 [math]\displaystyle{ f:[a,b]\to\mathbb{R} }[/math] can be defined by the relation

[math]\displaystyle{ \int_a^b f(x)\,dx = \lim_{\Delta x\to 0}\sum f(x_i)\,\Delta x, }[/math]

where the limit is taken over all partitions of the interval [math]\displaystyle{ [a,b] }[/math] 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

[math]\displaystyle{ \prod_a^b \big(1 + f(x)\,dx\big) = \lim_{\Delta x \to 0} \prod \big(1 + f(x_i)\,\Delta x\big). }[/math]

The type I product integral corresponds to Volterra's original definition.[2][14][15] The following relationship exists for scalar functions [math]\displaystyle{ f:[a,b] \to \mathbb{R} }[/math]:

[math]\displaystyle{ \prod_a^b \big(1 + f(x)\,dx\big) = \exp\left(\int_a^b f(x) \,dx\right), }[/math]

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 [16]

Left Product integral

[math]\displaystyle{ P(A,D)=\prod_{i=m}^{1}(\mathbb{1}+A(\xi_i)\Delta t_i) = (\mathbb{1}+A(\xi_m)\Delta t_m) \cdots (\mathbb{1}+A(\xi_1)\Delta t_1) }[/math]

With the notation of left products (i.e. normal products applied from left)

[math]\displaystyle{ \prod_a^b (\mathbb{1}+A(t)dt)=\lim_{\max \Delta t_i \to 0} P(A,D) }[/math]

Right Product Integral

[math]\displaystyle{ P(A,D)^*=\prod_{i=1}^{m}(\mathbb{1}+A(\xi_i)\Delta t_i) = (\mathbb{1}+A(\xi_1)\Delta t_1) \cdots (\mathbb{1}+A(\xi_m)\Delta t_m) }[/math]

With the notation of right products (i.e. applied from right)

[math]\displaystyle{ (\mathbb{1}+A(t)dt) \prod_a^b =\lim_{\max \Delta t_i \to 0} P(A,D)^* }[/math]

Where [math]\displaystyle{ \mathbb{1} }[/math] 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.[2]

Type II: geometric integral

[math]\displaystyle{ \prod_a^b f(x)^{dx} = \lim_{\Delta x \to 0} \prod{f(x_i)^{\Delta x}} = \exp\left(\int_a^b \ln f(x) \,dx\right), }[/math]

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

[math]\displaystyle{ \prod_{i=a}^b }[/math]

(with [math]\displaystyle{ i, a, b \in \mathbb{Z} }[/math]) and the multiplicative analog to the (normal/standard/additive) integral

[math]\displaystyle{ \int_a^b dx }[/math]

(with [math]\displaystyle{ x \in [a,b] }[/math]):

additive multiplicative
discrete [math]\displaystyle{ \sum_{i=a}^b f(i) }[/math] [math]\displaystyle{ \prod_{i=a}^b f(i) }[/math]
continuous [math]\displaystyle{ \int_a^b f(x)\,dx }[/math] [math]\displaystyle{ \prod_a^b f(x)^{dx} }[/math]

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:

[math]\displaystyle{ \ln \prod_a^b p(x)^{dx} = \int_a^b \ln p(x) \,dx. }[/math]

Type III: bigeometric integral

[math]\displaystyle{ \prod_a^b f(x)^{d(\ln x)} = \exp\left(\int_{\ln(a)}^{\ln(b)} \ln f(e^x) \,dx\right), }[/math]


The type III product integral is called the bigeometric integral and is a multiplicative operator.

Results

Basic results

The following results are for the type II product integral (the geometric integral). Other types produce other results.

[math]\displaystyle{ \prod_a^b c^{dx} = c^{b-a}, }[/math]
[math]\displaystyle{ \prod_a^b x^{dx} = \frac{b^b}{a^a} {\rm e}^{a-b}, }[/math]
[math]\displaystyle{ \prod_0^b x^{dx} = b^b {\rm e}^{-b}, }[/math]
[math]\displaystyle{ \prod_a^b \left(f(x)^k\right)^{dx} = \left(\prod_a^b f(x)^{dx}\right)^k, }[/math]
[math]\displaystyle{ \prod_a^b \left(c^{f(x)}\right)^{dx} = c^{\int_a^b f(x) \,dx}, }[/math]

The geometric integral (type II above) plays a central role in the geometric calculus,[3][4][17] which is a multiplicative calculus. The inverse of the geometric integral, which is the geometric derivative, denoted [math]\displaystyle{ f^*(x) }[/math], is defined using the following relationship:

[math]\displaystyle{ f^*(x)=\exp\left(\frac{f'(x)}{f(x)}\right) }[/math]

Thus, the following can be concluded:

The fundamental theorem
[math]\displaystyle{ \prod_a^b f^*(x)^{dx} = \prod_a^b \exp\left(\frac{f'(x)}{f(x)} \,dx\right) = \frac{f(b)}{f(a)}, }[/math]
Product rule
[math]\displaystyle{ (fg)^* = f^* g^*. }[/math]
Quotient rule
[math]\displaystyle{ (f/g)^* = f^*/g^*. }[/math]
Law of large numbers
[math]\displaystyle{ \sqrt[n]{X_1 X_2 \cdots X_n} \underset{n \to \infty}{\longrightarrow} \prod_x X^{dF(x)}, }[/math]

where X is a random variable with probability distribution F(x).

Compare with the standard law of large numbers:

[math]\displaystyle{ \frac{X_1 + X_2 + \cdots + X_n}{n} \underset{n \to \infty}{\longrightarrow} \int X \,dF(x). }[/math]

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 [math]\displaystyle{ f: [a,b] \to \mathbb{R} }[/math] with corresponding partition [math]\displaystyle{ a = y_0 \lt y_1 \lt \dots \lt y_m }[/math] and a tagged partition

[math]\displaystyle{ a = x_0 \lt x_1 \lt \dots \lt x_n = b, \quad x_0 \le t_0 \le x_1, x_1 \le t_1 \le x_2, \dots, x_{n-1} \le t_{n-1} \le x_n, }[/math]

one approximation of the "Riemann definition" of the type I product integral is given by[18]

[math]\displaystyle{ \prod_{k=0}^{n-1} \left[ \big(1 + f(t_k)\big) \cdot (x_{k+1} - x_k) \right]. }[/math]

The (type I) product integral was defined to be, roughly speaking, the limit of these products by Ludwig Schlesinger in a 1931 article.[which?]

Another approximation of the "Riemann definition" of the type I product integral is defined as

[math]\displaystyle{ \prod_{k=0}^{n-1} \exp\big(f(t_k) \cdot (x_{k+1} - x_k)\big). }[/math]

When [math]\displaystyle{ f }[/math] is a constant function, the limit of the first type of approximation is equal to the second type of approximation.[19] 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[20] for any product-integrable function [math]\displaystyle{ f }[/math], 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[21] the "Lebesgue (type I) product integral" of a step function as

[math]\displaystyle{ \prod_a^b \big(1 + f(x) \,dx\big) \overset{def}{=} \prod_{k=0}^{m-1} \exp\big(f(s_k) \cdot (y_{k+1} - y_k)\big), }[/math]

where [math]\displaystyle{ y_0 \lt a = s_0 \lt y_1 \lt \dots \lt y_{n-1} \lt s_{n-1} \lt y_n = b }[/math] is a tagged partition, and again [math]\displaystyle{ a = y_0 \lt y_1 \lt \dots \lt y_m }[/math] is the partition corresponding to the step function [math]\displaystyle{ f }[/math]. (In contrast, the corresponding quantity would not be unambiguously defined using the first type of approximation.)

This generalizes to arbitrary measure spaces readily. If [math]\displaystyle{ X }[/math] is a measure space with measure [math]\displaystyle{ \mu }[/math], then for any product-integrable simple function [math]\displaystyle{ f(x) = \sum_{k=1}^n a_k I_{A_k}(x) }[/math] (i.e. a conical combination of the indicator functions for some disjoint measurable sets [math]\displaystyle{ A_0, A_1, \dots, A_{m-1} \subseteq X }[/math]), its type I product integral is defined to be

[math]\displaystyle{ \prod_X \big(1 + f(x) \,d\mu(x)\big) \overset{def}{=} \prod_{k=0}^{m-1} \exp\big(a_k \mu(A_k)\big), }[/math]

since [math]\displaystyle{ a_k }[/math] is the value of [math]\displaystyle{ f }[/math] at any point of [math]\displaystyle{ A_k }[/math]. In the special case where [math]\displaystyle{ X = \mathbb{R} }[/math], [math]\displaystyle{ \mu }[/math] is Lebesgue measure, and all of the measurable sets [math]\displaystyle{ A_k }[/math] 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 Volterra product integral of any product-integrable function [math]\displaystyle{ f }[/math] 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 [math]\displaystyle{ f }[/math]:

[math]\displaystyle{ \ln \left(\prod_X \big(1 + f(x) \,d\mu(x)\big) \right) = \ln \left( \prod_{k=0}^{m-1} \exp\big(a_k \mu(A_k)\big) \right) = \sum_{k=0}^{m-1} a_k \mu(A_k) = \int_X f(x) \,d\mu(x) \iff }[/math]
[math]\displaystyle{ \prod_X \big(1 + f(x) \,d\mu(x)\big) = \exp \left( \int_X f(x) \,d\mu(x) \right), }[/math]

where we used the definition of integral for simple functions. Moreover, because continuous functions like [math]\displaystyle{ \exp }[/math] can be interchanged with limits, and the product integral of any product-integrable function [math]\displaystyle{ f }[/math] is equal to the limit of product integrals of simple functions, it follows that the relationship

[math]\displaystyle{ \prod_X \big(1 + f(x) \,d\mu(x)\big) = \exp \left( \int_X f(x) \,d\mu(x) \right) }[/math]

holds generally for any product-integrable [math]\displaystyle{ f }[/math]. This clearly generalizes the property mentioned above.

The Volterra product integral is multiplicative as a set function,[22] which can be shown using the above property. More specifically, given a product-integrable function [math]\displaystyle{ f }[/math] one can define a set function [math]\displaystyle{ {\cal V}_f }[/math] by defining, for every measurable set [math]\displaystyle{ B \subseteq X }[/math],

[math]\displaystyle{ {\cal V}_f(B) \overset{def}{=} \prod_B \big(1 + f(x) \,d\mu(x)\big) \overset{def}{=} \prod_X \big(1 + (f \cdot I_B)(x) \,d\mu(x)\big), }[/math]

where [math]\displaystyle{ I_B(x) }[/math] denotes the indicator function of [math]\displaystyle{ B }[/math]. Then for any two disjoint measurable sets [math]\displaystyle{ B_1, B_2 }[/math] one has

[math]\displaystyle{ \begin{align} {\cal V}_f(B_1 \sqcup B_2) &= \prod_{B_1 \sqcup B_2} \big(1 + f(x) \,d\mu(x)\big) \\ &= \exp\left( \int_{B_1 \sqcup B_2} f(x) \,d\mu(x) \right) \\ &= \exp\left( \int_{B_1} f(x) \,d\mu(x) + \int_{B_2} f(x) \,d\mu(x) \right) \\ &= \exp\left( \int_{B_1} f(x) \,d\mu(x) \right) \exp\left( \int_{B_2} f(x) \,d\mu(x) \right) \\ &= \prod_{B_1} (1 + f(x)d \mu(x)) \prod_{ B_2} (1 + f(x) \,d\mu(x)) \\ &= {\cal V}_f(B_1 ) {\cal V}_f(B_2). \end{align} }[/math]

This property can be contrasted with measures, which are additive set functions.

However the Volterra product integral is not multiplicative as a functional. Given two product-integrable functions [math]\displaystyle{ f , g }[/math], and a measurable set [math]\displaystyle{ A }[/math], it is generally the case that

[math]\displaystyle{ \prod_A \big(1 + (fg)(x) \,d\mu(x)\big) \neq \prod_A \big(1 + f(x) \,d\mu(x)\big) \prod_A \big(1 + g(x) \,d\mu(x)\big). }[/math]

Type II: geometric integral

If [math]\displaystyle{ X }[/math] is a measure space with measure [math]\displaystyle{ \mu }[/math], then for any product-integrable simple function [math]\displaystyle{ f(x) = \sum_{k=1}^n a_k I_{A_k}(x) }[/math] (i.e. a conical combination of the indicator functions for some disjoint measurable sets [math]\displaystyle{ A_0, A_1, \dots, A_{m-1} \subseteq X }[/math]), its type II product integral is defined to be

[math]\displaystyle{ \prod_X f(x)^{d\mu(x)} \overset{def}{=} \prod_{k=0}^{m-1} a_k^{\mu(A_k)}. }[/math]

This can be seen to generalize the definition given above.

Taking logarithms of both sides, we see that for any product-integrable simple function [math]\displaystyle{ f }[/math]:

[math]\displaystyle{ \ln \left( \prod_X f(x)^{d\mu(x)} \right) = \sum_{k=0}^{m-1} \ln(a_k) \mu(A_k) = \int_X \ln f(x) \,d\mu (x) \iff \prod_X f(x)^{d\mu(x)} = \exp\left( \int_X \ln f(x) \,d\mu (x) \right), }[/math]

where we have used the definition of the Lebesgue integral for simple functions. This observation, analogous to the one already made above, allows one to entirely reduce the "Lebesgue theory of geometric integrals" to the Lebesgue theory of (classical) integrals. In other words, because continuous functions like [math]\displaystyle{ \exp }[/math] and [math]\displaystyle{ \ln }[/math] can be interchanged with limits, and the product integral of any product-integrable function [math]\displaystyle{ f }[/math] is equal to the limit of some increasing sequence of product integrals of simple functions, it follows that the relationship

[math]\displaystyle{ \prod_X f(x)^{d\mu(x)} = \exp\left( \int_X \ln f(x) \,d\mu(x) \right) }[/math]

holds generally for any product-integrable [math]\displaystyle{ f }[/math]. This generalizes the property of geometric integrals mentioned above.

See also

References

  1. V. Volterra, B. Hostinský, Opérations Infinitésimales Linéaires, Gauthier-Villars, Paris (1938).
  2. 2.0 2.1 2.2 A. Slavík, Product integration, its history and applications, ISBN 80-7378-006-2, Matfyzpress, Prague, 2007.
  3. 3.0 3.1 3.2 M. Grossman, R. Katz, Non-Newtonian Calculus, ISBN 0-912938-01-3, Lee Press, 1972.
  4. 4.0 4.1 Michael Grossman. The First Nonlinear System of Differential And Integral Calculus, ISBN:0977117006, 1979.
  5. 5.0 5.1 Michael Grossman. Bigeometric Calculus: A System with a Scale-Free Derivative, ISBN:0977117030, 1983.
  6. Luc Florack and Hans van Assen."Multiplicative calculus in biomedical image analysis", Journal of Mathematical Imaging and Vision, doi:10.1007/s10851-011-0275-1, 2011.
  7. Diana Andrada Filip and Cyrille Piatecki. "An overview on non-Newtonian calculus and its potential applications to economics", Applied Mathematics – A Journal of Chinese Universities, Volume 28, China Society for Industrial and Applied Mathematics, Springer, 2014.
  8. Agamirza E. Bashirov, Emine Misirli, Yucel Tandogdu, and Ali Ozyapici."On modelling with multiplicative differential equations", Applied Mathematics – A Journal of Chinese Universities, Volume 26, Number 4, pages 425–428, doi:10.1007/s11766-011-2767-6, Springer, 2011.
  9. Marek Rybaczuk."Critical growth of fractal patterns in biological systems", Acta of Bioengineering and Biomechanics, Volume 1, Number 1, Wroclaw University of Technology, 1999.
  10. Marek Rybaczuk, Alicja Kedzia and Witold Zielinski (2001) "The concept of physical and fractal dimension II. The differential calculus in dimensional spaces", Chaos, Solitons, & FractalsVolume 12, Issue 13, October 2001, pages 2537–2552.
  11. Aniszewska, Dorota (October 2007). "Multiplicative Runge–Kutta methods". Nonlinear Dynamics 50 (1–2): 265–272. doi:10.1007/s11071-006-9156-3. 
  12. Dorota Aniszewska and Marek Rybaczuk (2005) "Analysis of the multiplicative Lorenz system", Chaos, Solitons & Fractals Volume 25, Issue 1, July 2005, pages 79–90.
  13. Fernando Córdova-Lepe. "The multiplicative derivative as a measure of elasticity in economics", TMAT Revista Latinoamericana de Ciencias e Ingeniería, Volume 2, Number 3, 2006.
  14. Dollard, J. D.; Friedman, C. N. (1979). Product integration with applications to differential equations. Addison Wesley. ISBN 0-201-13509-4. 
  15. Gantmacher, F. R. (1959). The Theory of Matrices. 1 and 2. 
  16. Cherednikov, Igor Olegovich; Mertens, Tom; Van der Veken, Frederik (2 December 2019). Wilson Lines in Quantum Field Theory. ISBN 9783110651690. https://books.google.com/books?id=hUDEDwAAQBAJ&pg=PA64. 
  17. A. E. Bashirov, E. M. Kurpınar, A. Özyapıcı. Multiplicative calculus and its applications, Journal of Mathematical Analysis and Applications, 2008.
  18. A. Slavík, Product integration, its history and applications, p. 65. Matfyzpress, Prague, 2007. ISBN 80-7378-006-2.
  19. A. Slavík, Product integration, its history and applications, p. 71. Matfyzpress, Prague, 2007. ISBN 80-7378-006-2.
  20. A. Slavík, Product integration, its history and applications, p. 72. Matfyzpress, Prague, 2007. ISBN 80-7378-006-2.
  21. A. Slavík, Product integration, its history and applications, p. 80. Matfyzpress, Prague, 2007. ISBN 80-7378-006-2
  22. Gill, Richard D., Soren Johansen. "A Survey of Product Integration with a View Toward Application in Survival Analysis". The Annals of Statistics 18, no. 4 (December 1990): 1501—555, p. 1503.

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