On Approximation of Copulas, Solution of Elliptic Problem, Using the Finite Difference Method: Generalization

Remi Guillaume Bagré; Frédéric Béré; Kpèbbèwèrè Cédric Somé; Paliguiwindé Dieudonné Ibrango

doi:doi:10.11648/j.ajtas.20211006.15

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On Approximation of Copulas, Solution of Elliptic Problem, Using the Finite Difference Method: Generalization

Remi Guillaume Bagré, Frédéric Béré, Kpèbbèwèrè Cédric Somé, Paliguiwindé Dieudonné Ibrango

Published in American Journal of Theoretical and Applied Statistics (Volume 10, Issue 6)

Received: 26 October 2021 Accepted: 26 November 2021 Published: 9 December 2021

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Abstract

In this paper, we generalize a copula construction method discussed in one of our papers. For this purpose we consider the general form of a linear elliptic PDE. Indeed, a physical interpretation of elliptic equations comes from the notion of conservative flow given by a gradient. This notion provides a mathematical model for equilibrium conservation laws in linear behaviour. This can be applied to many areas of science. Thus, the aim of this paper is to construct a new class of bivariate copulas by solving an elliptic partial differential equation with a Dirichlet condition at the boundary. Copulas belonging to this class allow us to study the stochastic behaviour of the notion of conservative flows. In other words, these copulas will allow us to have an idea on the dependence of those physical phenomena which are governed by elliptic PDEs. For this purpose, we use a discretization method which is the finite difference method which is a common technique for finding approximate solutions of partial differential equations that consists in solving a system of relations (numerical scheme) connecting the values of the unknown functions at some points sufficiently close to each other. For the finite difference method, a mesh is made. This is a set of isolated points called nodes located in the domain of definition of the functions subject to the partial differential equations, a grid on which only the nodes of which the unknowns corresponding to the approximate values of these functions are defined. The mesh also includes nodes located on the boundary of the domain (or at least "close" to this boundary) in order to be able to impose the boundary conditions and/or the initial condition with sufficient accuracy. The primary quality of a mesh is to cover the domain in which it develops as well as possible, to limit the distance between each node and its nearest neighbour. However, the mesh must also allow the discrete formulation of the differentiation operators to be expressed: for this reason, the nodes of the mesh are most often located on a grid whose main directions are the axes of the variables. In the main results of this paper (see section 3), we give a discretization of the solution of the problem followed by a simulation with the MATLAB software of this approximated solution and presenting the discretization errors.

Published in	American Journal of Theoretical and Applied Statistics (Volume 10, Issue 6)
DOI	10.11648/j.ajtas.20211006.15
Page(s)	257-261
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Copulas, Linear Elliptic PDE, Boundary Value Problem, Finite Difference Method, Discretization

References

[1]	Adams R. A., J. J. F. Fournier (2003), Sobolev Spaces. Oxford, Elsevier LTD, Academic Press.
[2]	Chervenov, Nikolay; Iordanov Iordan; Kostadinov Boyan (2018), n-dimensional copulas and weak derivatives. Serdica Mathematical Journal. 2018, Vol. 44 Issue 3/4, p 413-438. 26p.
[3]	Karl Friedrich Siburg and Pavel A. Stoimenov (2008), A scalar product for copulas. J. Math. Anal. Appl. 344 (2008) 429-439.
[4]	Gu Yan, Sun Hong Guang, A meshless method for solving three-dimensional time fractional difusion equation with variable-order derivatives, Applied Mathematical Modelling, ELSEVIER Volume 78, February 2020, Pages 539-549.
[5]	Hao Xia, Yan Gu Generalized finite difference method for electroelastic analysis of three-dimensional piezoelectric structures, Applied Mathematics Letters, ELSEVIER Volume 117, July 2021, 107084.
[6]	Iordan Iordanov, Nikolay Chervenov (2015), Copulas on sobolev spaces. Comptes rendus de l’Académie bulgare des Sciences Tome 68, No 1, 2015.
[7]	Nagumo M. (1967), Lecture notes on modern theory of partial diferential equations, Moscow, Mir (in Russian). Academic Press.
[8]	Naoyuki Ishimura, Yasukazu Yoshizawa (2012), Evolution of multivariate copulas in discrete processes. Graduate School of Economics, Hitotsubashi University, Kunitachi, Tokyo 186-8601, Japan.
[9]	R. B. Nelsen An introduction to copulas, vol. 139 of Springer Series in Statistics, Springer, New York, NY, USA, 2^nd edition, 1999.
[10]	Remi Guillaume Bagré, Frédéric Béré and Vini Yves Bernadin Loyara, Construction of a Class of Copula Using the Finite Difference Method. Hindawi, Journal of Function Spaces, Volume 2021, Article ID 5271105, 8 pages.
[11]	Remi guillaume Bagré, Frédéric Béré and Abdoulaye Compaoré, finite element method used to approximate bivariate Copulas with non-homogeneous condition. Advances in Differential Equations and Control Processes, Volume 25, Number 2, 2021, Pages 231-243.
[12]	Remi Guillaume Bagré, Vini Yves Bernadin Loyara and Diakarya Barro, Spatial characterization of stochastic dependence using copulas. Far East Journal of Theoretical Statistics. Volume 58, Number 1, 2020, Pages 21-35.
[13]	Vincent Manet (2012), La Méthode des éléments Finis: Vulgarisation des aspects Mathématiques, Illustration des capacités de la méthode HAL-Archives ouvertes.
[14]	Yasukazu Yoshizawa and Naoyuki Ishimura (2011), Evolution of bivariate copulas in discrete processes. JSIAM Letters Vol. 3 (2011) pp. 77, Japan Society for Industrial and Applied Mathematics.
[15]	WILLIAM F. DARSOW and ELWOOD T. OLSEN (1995), Norm for Copulas. Internat. J. Math. and Math. Sci. VOL. 18 NO. 3 (1995) 417-436.

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Remi Guillaume Bagré, Frédéric Béré, Kpèbbèwèrè Cédric Somé, Paliguiwindé Dieudonné Ibrango. (2021). On Approximation of Copulas, Solution of Elliptic Problem, Using the Finite Difference Method: Generalization. American Journal of Theoretical and Applied Statistics, 10(6), 257-261. https://doi.org/10.11648/j.ajtas.20211006.15

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Remi Guillaume Bagré; Frédéric Béré; Kpèbbèwèrè Cédric Somé; Paliguiwindé Dieudonné Ibrango. On Approximation of Copulas, Solution of Elliptic Problem, Using the Finite Difference Method: Generalization. Am. J. Theor. Appl. Stat. 2021, 10(6), 257-261. doi: 10.11648/j.ajtas.20211006.15

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AMA Style

Remi Guillaume Bagré, Frédéric Béré, Kpèbbèwèrè Cédric Somé, Paliguiwindé Dieudonné Ibrango. On Approximation of Copulas, Solution of Elliptic Problem, Using the Finite Difference Method: Generalization. Am J Theor Appl Stat. 2021;10(6):257-261. doi: 10.11648/j.ajtas.20211006.15

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@article{10.11648/j.ajtas.20211006.15,
  author = {Remi Guillaume Bagré and Frédéric Béré and Kpèbbèwèrè Cédric Somé and Paliguiwindé Dieudonné Ibrango},
  title = {On Approximation of Copulas, Solution of Elliptic Problem, Using the Finite Difference Method: Generalization},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {10},
  number = {6},
  pages = {257-261},
  doi = {10.11648/j.ajtas.20211006.15},
  url = {https://doi.org/10.11648/j.ajtas.20211006.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20211006.15},
  abstract = {In this paper, we generalize a copula construction method discussed in one of our papers. For this purpose we consider the general form of a linear elliptic PDE. Indeed, a physical interpretation of elliptic equations comes from the notion of conservative flow given by a gradient. This notion provides a mathematical model for equilibrium conservation laws in linear behaviour. This can be applied to many areas of science. Thus, the aim of this paper is to construct a new class of bivariate copulas by solving an elliptic partial differential equation with a Dirichlet condition at the boundary. Copulas belonging to this class allow us to study the stochastic behaviour of the notion of conservative flows. In other words, these copulas will allow us to have an idea on the dependence of those physical phenomena which are governed by elliptic PDEs. For this purpose, we use a discretization method which is the finite difference method which is a common technique for finding approximate solutions of partial differential equations that consists in solving a system of relations (numerical scheme) connecting the values of the unknown functions at some points sufficiently close to each other. For the finite difference method, a mesh is made. This is a set of isolated points called nodes located in the domain of definition of the functions subject to the partial differential equations, a grid on which only the nodes of which the unknowns corresponding to the approximate values of these functions are defined. The mesh also includes nodes located on the boundary of the domain (or at least "close" to this boundary) in order to be able to impose the boundary conditions and/or the initial condition with sufficient accuracy. The primary quality of a mesh is to cover the domain in which it develops as well as possible, to limit the distance between each node and its nearest neighbour. However, the mesh must also allow the discrete formulation of the differentiation operators to be expressed: for this reason, the nodes of the mesh are most often located on a grid whose main directions are the axes of the variables. In the main results of this paper (see section 3), we give a discretization of the solution of the problem followed by a simulation with the MATLAB software of this approximated solution and presenting the discretization errors.},
 year = {2021}
}

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TY  - JOUR
T1  - On Approximation of Copulas, Solution of Elliptic Problem, Using the Finite Difference Method: Generalization
AU  - Remi Guillaume Bagré
AU  - Frédéric Béré
AU  - Kpèbbèwèrè Cédric Somé
AU  - Paliguiwindé Dieudonné Ibrango
Y1  - 2021/12/09
PY  - 2021
N1  - https://doi.org/10.11648/j.ajtas.20211006.15
DO  - 10.11648/j.ajtas.20211006.15
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 257
EP  - 261
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20211006.15
AB  - In this paper, we generalize a copula construction method discussed in one of our papers. For this purpose we consider the general form of a linear elliptic PDE. Indeed, a physical interpretation of elliptic equations comes from the notion of conservative flow given by a gradient. This notion provides a mathematical model for equilibrium conservation laws in linear behaviour. This can be applied to many areas of science. Thus, the aim of this paper is to construct a new class of bivariate copulas by solving an elliptic partial differential equation with a Dirichlet condition at the boundary. Copulas belonging to this class allow us to study the stochastic behaviour of the notion of conservative flows. In other words, these copulas will allow us to have an idea on the dependence of those physical phenomena which are governed by elliptic PDEs. For this purpose, we use a discretization method which is the finite difference method which is a common technique for finding approximate solutions of partial differential equations that consists in solving a system of relations (numerical scheme) connecting the values of the unknown functions at some points sufficiently close to each other. For the finite difference method, a mesh is made. This is a set of isolated points called nodes located in the domain of definition of the functions subject to the partial differential equations, a grid on which only the nodes of which the unknowns corresponding to the approximate values of these functions are defined. The mesh also includes nodes located on the boundary of the domain (or at least "close" to this boundary) in order to be able to impose the boundary conditions and/or the initial condition with sufficient accuracy. The primary quality of a mesh is to cover the domain in which it develops as well as possible, to limit the distance between each node and its nearest neighbour. However, the mesh must also allow the discrete formulation of the differentiation operators to be expressed: for this reason, the nodes of the mesh are most often located on a grid whose main directions are the axes of the variables. In the main results of this paper (see section 3), we give a discretization of the solution of the problem followed by a simulation with the MATLAB software of this approximated solution and presenting the discretization errors.
VL  - 10
IS  - 6
ER  -

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Author Information

Remi Guillaume Bagré

Departement of Mathematics, Norbert ZONGO University, Koudougou, Burkina Faso
Frédéric Béré

Departement of Mathematics, Institute of Sciences, Ouagadougou, Burkina Faso
Kpèbbèwèrè Cédric Somé

Departement of Mathematics, Virtual University of Burkina Faso, Ouagadougou, Burkina Faso
Paliguiwindé Dieudonné Ibrango

Departement of Mathematics, Joseph Ki-ZERBO University, Ouagadougou, Burkina Faso

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APA Style

Remi Guillaume Bagré, Frédéric Béré, Kpèbbèwèrè Cédric Somé, Paliguiwindé Dieudonné Ibrango. (2021). On Approximation of Copulas, Solution of Elliptic Problem, Using the Finite Difference Method: Generalization. American Journal of Theoretical and Applied Statistics, 10(6), 257-261. https://doi.org/10.11648/j.ajtas.20211006.15

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ACS Style

Remi Guillaume Bagré; Frédéric Béré; Kpèbbèwèrè Cédric Somé; Paliguiwindé Dieudonné Ibrango. On Approximation of Copulas, Solution of Elliptic Problem, Using the Finite Difference Method: Generalization. Am. J. Theor. Appl. Stat. 2021, 10(6), 257-261. doi: 10.11648/j.ajtas.20211006.15

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AMA Style

Remi Guillaume Bagré, Frédéric Béré, Kpèbbèwèrè Cédric Somé, Paliguiwindé Dieudonné Ibrango. On Approximation of Copulas, Solution of Elliptic Problem, Using the Finite Difference Method: Generalization. Am J Theor Appl Stat. 2021;10(6):257-261. doi: 10.11648/j.ajtas.20211006.15

Copy | Download

@article{10.11648/j.ajtas.20211006.15,
  author = {Remi Guillaume Bagré and Frédéric Béré and Kpèbbèwèrè Cédric Somé and Paliguiwindé Dieudonné Ibrango},
  title = {On Approximation of Copulas, Solution of Elliptic Problem, Using the Finite Difference Method: Generalization},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {10},
  number = {6},
  pages = {257-261},
  doi = {10.11648/j.ajtas.20211006.15},
  url = {https://doi.org/10.11648/j.ajtas.20211006.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20211006.15},
  abstract = {In this paper, we generalize a copula construction method discussed in one of our papers. For this purpose we consider the general form of a linear elliptic PDE. Indeed, a physical interpretation of elliptic equations comes from the notion of conservative flow given by a gradient. This notion provides a mathematical model for equilibrium conservation laws in linear behaviour. This can be applied to many areas of science. Thus, the aim of this paper is to construct a new class of bivariate copulas by solving an elliptic partial differential equation with a Dirichlet condition at the boundary. Copulas belonging to this class allow us to study the stochastic behaviour of the notion of conservative flows. In other words, these copulas will allow us to have an idea on the dependence of those physical phenomena which are governed by elliptic PDEs. For this purpose, we use a discretization method which is the finite difference method which is a common technique for finding approximate solutions of partial differential equations that consists in solving a system of relations (numerical scheme) connecting the values of the unknown functions at some points sufficiently close to each other. For the finite difference method, a mesh is made. This is a set of isolated points called nodes located in the domain of definition of the functions subject to the partial differential equations, a grid on which only the nodes of which the unknowns corresponding to the approximate values of these functions are defined. The mesh also includes nodes located on the boundary of the domain (or at least "close" to this boundary) in order to be able to impose the boundary conditions and/or the initial condition with sufficient accuracy. The primary quality of a mesh is to cover the domain in which it develops as well as possible, to limit the distance between each node and its nearest neighbour. However, the mesh must also allow the discrete formulation of the differentiation operators to be expressed: for this reason, the nodes of the mesh are most often located on a grid whose main directions are the axes of the variables. In the main results of this paper (see section 3), we give a discretization of the solution of the problem followed by a simulation with the MATLAB software of this approximated solution and presenting the discretization errors.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - On Approximation of Copulas, Solution of Elliptic Problem, Using the Finite Difference Method: Generalization
AU  - Remi Guillaume Bagré
AU  - Frédéric Béré
AU  - Kpèbbèwèrè Cédric Somé
AU  - Paliguiwindé Dieudonné Ibrango
Y1  - 2021/12/09
PY  - 2021
N1  - https://doi.org/10.11648/j.ajtas.20211006.15
DO  - 10.11648/j.ajtas.20211006.15
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 257
EP  - 261
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20211006.15
AB  - In this paper, we generalize a copula construction method discussed in one of our papers. For this purpose we consider the general form of a linear elliptic PDE. Indeed, a physical interpretation of elliptic equations comes from the notion of conservative flow given by a gradient. This notion provides a mathematical model for equilibrium conservation laws in linear behaviour. This can be applied to many areas of science. Thus, the aim of this paper is to construct a new class of bivariate copulas by solving an elliptic partial differential equation with a Dirichlet condition at the boundary. Copulas belonging to this class allow us to study the stochastic behaviour of the notion of conservative flows. In other words, these copulas will allow us to have an idea on the dependence of those physical phenomena which are governed by elliptic PDEs. For this purpose, we use a discretization method which is the finite difference method which is a common technique for finding approximate solutions of partial differential equations that consists in solving a system of relations (numerical scheme) connecting the values of the unknown functions at some points sufficiently close to each other. For the finite difference method, a mesh is made. This is a set of isolated points called nodes located in the domain of definition of the functions subject to the partial differential equations, a grid on which only the nodes of which the unknowns corresponding to the approximate values of these functions are defined. The mesh also includes nodes located on the boundary of the domain (or at least "close" to this boundary) in order to be able to impose the boundary conditions and/or the initial condition with sufficient accuracy. The primary quality of a mesh is to cover the domain in which it develops as well as possible, to limit the distance between each node and its nearest neighbour. However, the mesh must also allow the discrete formulation of the differentiation operators to be expressed: for this reason, the nodes of the mesh are most often located on a grid whose main directions are the axes of the variables. In the main results of this paper (see section 3), we give a discretization of the solution of the problem followed by a simulation with the MATLAB software of this approximated solution and presenting the discretization errors.
VL  - 10
IS  - 6
ER  -

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