Evolution of multivariate copulas in continuous and discrete processes

• Published date: 01 January 2018
• Author: Naoyuki Ishimura,Yasukazu Yoshizawa
• Date: 01 January 2018
• DOI: http://doi.org/10.1002/isaf.1420

RESEARCH ARTICLE

Evolution of multivariate copulas in continuous and discrete

processes

Yasukazu Yoshizawa

1

|Naoyuki Ishimura

2

1

Toyo University, Tokyo, Japan

2

Chuo University, Tokyo, Japan

Correspondence

Yasukazu Yoshizawa, 5‐28‐20 Hakusan

Bunkyo‐ku Tokyo, 112‐8606, Japan.

Email: Yoshizawa@toyo.jp

Funding information

Japan Society for the Promotion of Science

(JSPS), Grant/Award Numbers: 15K04992,

Grant‐in‐Aid for Scientific Research (C)

(kakenhi)

JEL Classification: C31; C32; C46; C53; C58;

G22

Summary

There has been much interest in copulas, which are known to provide a flexible tool for analyzing

the dependence structure among random variables. Dependence relations must be dynamic

rather than static in nature. However, copulas are useful mainly for static matters. Thus we intro-

duce evolving multivariate copulas, which transform through time autonomously governed by the

multivariate heat equation. Our aims are to prove their existences and solutions to analyze their

transitions. Moreover, we construct discrete type to apply empirical data analysis and investigate

their properties, and prove that they converge to their original continuous type.

KEYWORDS

copula, partial differential equation, quantitative riskmanagement

1|INTRODUCTION

Dependence relations among random variables are one of the most

important aspects of probability and statistics. Analyzing dependence

structures is crucial from both theoretical and applied viewpoints.

Recently, members of the financial sector, such as insurance compa-

nies and their regulators, have recognized that it is critical to manage

these risks in a sophisticated manner, that is, quantitatively. Quantita-

tively measured risks play a central role in this management frame-

work. These entities face many kinds of risks, and the relations

between them are very complicated. Thus, it is crucial to reflect on

dependence relations to measure risks quantitatively: the more

dependence there is among risks, the less aggregated the risks are.

(see Yoshizawa, 2009, 2010).

Linear correlation is often recognized as a satisfactory measure of

dependence in risk management. However, it cannot capture the

non‐linear dependence relations that exist among many risk factors.

(see Embrechts, Lindskog, & McNeil, 2003; Embrechts, McNeil, &

Straumann, 1999; Embrechts, McNeil, & Straumann, 2002). What

expresses the dependence relations among risk factors? If we capture

a multivariate joint distribution of all the risk factors, we can recognize

their dependence structure probabilistically or statistically.

For this reason, there has been much interest in copulas. A copula

links multivariate joint distribution and univariate marginal distribu-

tions. Copulas are often employed to investigate the dependence

structure among random variables. The fundamental theorem of cop-

ulas, “Sklar's theorem”, first appeared in 1959. (see Schweizer & Sklar,

1983; Sklar, 1973). This elegant theorem claims that all multivariate

distributions have copulas, and that copulas are used in conjunction

with univariate distributions to construct multivariate distributions.

As for the study of copulas, Frees and Valdez (1998) and Tsukahara

(2003) provided useful introductive reviews. Nelsen (2006) wrote the

excellent standard textbook, providing a systematic development of

the theory of copulas, particularly bivariate copulas. The book by

Durante and Sempi (2016) and the books by Joe (1997, 2015) are also

helpful, and they cover multivariate copulas. Additionally, McNeil,

Frey, and Embrechts (2005) introduced copulas theoretically and prac-

tically, examining quantitative risk management for financial sectors.

Because of their flexibility, copulas have been extensively studied

and applied in a wide range of areas involving dependence relations.

(see Breymann, Dias, & Embrechts, 2003; Genest & MacKay, 1986;

Goda, 2010; Goda & Atkinson, 2009; Goorbergh, van den Genest, &

Werker, 2005; Lebrun & Dutfoy, 2009).

However, copulas are useful mainly for static matters; their defini-

tions themselves do not contain time variables. Mikosch (2005) sug-

gests, “Copulas do not fit into the existing framework of stochastic

processes and time series analysis; they are essentially static models

and are not useful for modeling dependence through time”. Nonethe-

less, a few exceptions exist: copulas and the Markov process, as in

Darsow, Nguyen, and Olsen (1992), and dynamic copulas, as in Patton

(2006). Copulas and Markov processes can be used to analyze the

dependence relations between Markov processes at different times.

Dynamic copulas involve the development of dynamic time series

models for financial return data using conditional copulas. Fermanian

and Wegkamp (2012) introduced the concept of pseudo‐copulas,

extending Patton's definition of conditional copulas. Hafner and

Received: 13 April 2017 Revised: 27 October 2017 Accepted: 21 December 2017

DOI: 10.1002/isaf.1420

44 Copyright © 2018 John Wiley & Sons, Ltd. Intell Sys Acc Fin Mgmt. 2018;25:44–59.wileyonlinelibrary.com/journal/isaf

Manner (2012), Härdle, Okhrin, and Wang (2010), and Manner and

Reznikova (2012) are excellent articles in this regard. Furthermore,

Kallsen and Tankov (2006) propose the Lévy copula for the Lévy pro-

cess, and a survey by Bielecki, Jakubowski, and Nieweglowski (2010)

introduced some copulas for stochastic processes.

The above previous dynamic copulas theories adopted parametric

movement of copula functions, such as Clayton copulas, Gaussian cop-

ulas, and t‐copulas. All of these theories predicted future relations

from past results and tendencies. We wish to find dynamic copulas

that transform autonomously governed by some laws such as evolu-

tion equations.

It is well known that rank correlations, which are defined as the

probability of concordance minus the probability of dis‐concordance

for a pair of distributed random vectors, are the prevailing measures

of dependence. Moreover these rank correlations are derived only

by copulas. That is to say, copulas can determine rank correlations.

Thus, we believe that it is natural to analyze only copulas in the study

of transformations of dependence structures through time. As a first

step, we start by investigating how copulas transform, if they evolve

in accordance with the heat equation, which is one of the basic partial

differential equations used to describe dynamic movements. We name

these transformations of copulas as “evolution of copulas”, where the

evolution comes from the evolution equation interpreted as the

differential law of the development (evolution) in time of a system.

In the case of bivariate variables, we already studied them as the

bivariate evolution of copulas in Ishimura and Yoshizawa (2011a, b),

Yoshizawa (2015), and Yoshizawa and Ishimura (2011a, b). Main con-

tribution of our study is that for any initial copulas their evolution of

copulas converges to the product copula and their rank correlations

converge to zero, where these results express the independent rela-

tions among variables

However, the objects of our study do not always consist of two

factor events, but usually multi‐factor events. Therefore multivariate

copulas must be necessary for any study and any application both the-

oretically and practically. There are some excellent books on multivar-

iate copulas, such as Durante and Sempi (2016), Joe (1997, 2015) and

McNeil, Frey and Embrechts (2005).

In this article we extend the theory from bivariate copulas to

multivariate ones, and study the evolution of multivariate copulas

governed by a multivariate heat equation. In conclusion, the evolution

of multivariate copulas have the same properties as the evolution of

bivariate copulas both in continuous processes and discrete processes.

Because of these results, we can apply these theories not only to two

variable cases but also to multi variable cases. The major contributions

of this article are that we can apply these theorems not only to the

events consisting of two factors, but also any phenomena.

We explain the details of evolution of multivariate copulas in

continuous processes in section 2, and those for discrete processes

in section 3. In expanding bivariate cases to multivariate cases, there

were some difficulties in proving their theorems in multivariate cases

compared to simple bivariate cases. Therefore we use some new

techniques for these proofs. Recently vine copulas have become

popular owning to their convenience in creating multivariate distri-

butions using only marginal distributions and their bivariate condi-

tional copulas. We study the outline of the evolution of vine

copulas in section 2.5. In section 4, we apply the evolution of multi-

variate copulas to empirical data. Finally, in section 5, we discuss the

prospects of evolution of copulas, as well as the concept of general-

ized evolution of copulas which is the extension type of discrete

evolution of copulas.

In this section 1, as an introduction, we summarize the basic

concept of copulas in section 1.1, the basic concept of rank corre-

lations in section 1.2, the continuous evolution of bivariate copulas

in section 1.3, and the discrete evolution of bivariate copulas in

section 1.4 so as to enable readers to understand multivariate evo-

lution of copulas easily, because bivariate cases are simpler than

multivariate cases.

1.1 |Bivariate copulas

We marshal the basic concepts and properties of bivariate copulas for

the preparation of the subsequent sections. We illustrate the definition

of bivariate copulas; describe Sklar's theorem, which plays a central

role in copula theories. For background issues in this section, we refer

mainly to McNeil, Frey and Embrechts (2005) and Nelsen (2006).

The bivariate copulas are defined by some conditions described in

the following (1.1) and (1.2). The property (1.1) is called 2‐increasing

condition, and the properties (1.2) are boundary conditions. The image

of a copula is shown diagrammatically in Figure 1.

1.1.1 |Copulas

Bivariate copula is a function C(u,v) from I

2

to I, which is defined by the

following conditions

2‐increasing condition;

Cu

2;v2

ðÞ−Cu

2;v1

ðÞ−Cu

1;v2

ðÞþCu

1;v1

ðÞ≥0;

for ui;vj



∈I2i;j¼1;2ðÞ;u1≤u2;v1≤v2:(1:1)

Boundary conditions;

Cu;0ðÞ¼C0;vðÞ¼0;

Cu;1ðÞ¼uand C1;vðÞ¼v:(1:2)

The following Sklar's theorem is the core theory among various

copula theories. Thanks to this theorem we can construct multivariate

distribution by coupling univariate marginal distributions. The Figure 2

is the conceptual image of the coupling using Sklar's theorem.

FIGURE 1 Copula

YOSHIZAWA AND ISHIMURA 45