Generalized evolution of bivariate copulas in discrete processes
Yasukazu YOSHIZAWA 1,* and Naoyuki ISHIMURA 2
1
2

Tokio Marine & Nichido Life, Tokyo 100-0005, Japan; yasukazu.yoshizawa@tmnf.jp
Chuo University, Tokyo 192-0351, Japan; naoyuki@tamacc.chuo-u.ac.jp

Abstract: There has been much interest in copulas, which are known to provide a flexible tool for analyzing thedependence structure among random variables. Dependence relations must be dynamic rather than static in nature.
However, copulas are useful mainly for static matters; their definitions themselves do not contain time variables. Thus
we introduce evolving copulas, which transform through time autonomously as governed by recurrence relation.
Keywords: Quantitative Risk Management; Copula; Recurrence relation

1. Introduction
Dependence relations among random variables are one of the most important subjects for probability and statistics
study. Analyzing dependence structures is crucial from both theoretical and applied viewpoints. Recently, members of
the financial sector, like insurance companies and their regulators, have recognized that it is critical to manage these risks
in a sophisticated way, that is, quantitatively. Quantitatively measured risks play a central role in this management
framework. These entities face many kinds of risks, and the relations between them are very complicated. Thus, it isimportant to reflect the dependence relations to measure the risks quantitatively: the more dependence there is amongrisks, the less aggregated the risks are. See Yoshizawa, Y. [19], [20].
Linear correlation is often recognized as a satisfactory measure of dependence in risk management. However, itcannot capture the non-linear dependence relations that exist among many risk factors. See Embrechits, P. et al. [3].
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 a copula. A copula links multivariate joint distribution
and univariate marginal distributions. Copulas are often employed to investigate the dependence structure
among random variables. The fundamental theorem of copulas, “Sklar’s theorem,” first appeared in 1959. See
Schweizer, B., & Sklar, A. [15], and Sklar, A. [16]. This elegant theorem claims that all multivariate
distributions have copulas, and that copulas are used in conjunction with univariate distributions to constructmultivariate distribution. As for the study of copulas, Free, E.W. et al. [4] and Tsukahara, H. [18] provide
useful introductive reviews. Nelsen, R.B. [14] wrote the excellent standard textbook, providing a systematicdevelopment of the theory of copulas, particularly bivariate copulas. Moreover, McNeil, A. J. et al. [12] introduce
copulas theoretically and practically, examining quantitative risk management for financial sectors. Copulas have been
extensively studied and applied in a wide range of areas concerning dependence relations, because of their flexibility. See
Breymann, W. et al. [1], Genest, G. et al. [5], and Goorbergh, R.W.J. et al. [6].We summarize the basic concept of
copulas and rank correlations in Section 2.
Copulas are useful mainly for static matters; their definitions themselves do not contain time variables. Mikosch, T.
[13] suggests, “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”. However a few exceptions exist:
copulas and the markov process, as in Darsow, W.F. et al. [2], and dynamic copulas, as in Patton, A.J. [17]. In thesearticles copulas and Markov processes can be used to analyze the dependence relations between markov processes atdifferent times, and dynamic copulas involve the development of dynamic time series models for financial return datausing conditional copulas.
It is well known that rank correlations, one of the prevailing measures of dependence, are derived only by copulas.
That is to say, copulas determine rank correlations. Therefore, it is natural to analyze only copulas in the study oftransformations of dependence structures through time. As a first step, we start to investigate how copulas transform, andif they evolve in accordance with the heat equation, which is one of the basic partial differential equations used to1

describe dynamic movements. We already studied the evolution of bivariate copulas in continuous and discrete processes,
and summarize them in Section 3.
However the evolution of copulas is not versatile; it is restricted to events whose essential dependences do notfluctuate but transform monotonically. The evolution of copulas fits with events where dependence decreases. In thispaper, we extend the evolution of bivariate copulas in discrete processes to the generalized evolution of copulas, which isgoverned by past copulas asymmetrically weighted by coefficients. We propose a prototype of the generalized evolutionof bivariate copulas in Section 4, and comment on their expected collaboration with Artificial Intelligence (AI) in Section
5. We hope the generalized evolution of copulas fits into various transformations of dependence structures, such as
rapidly strengthening dependences, becoming independent smoothly and so on.

2. Copulas and rank correlation
In this section, as an introduction, we summarize basic concept of copulas and rank correlations.
2.1 Basic concept of 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.
The bivariate copulas are defined by some conditions described in the following (1) and (2). The property (1) is called
2-increasing condition, and the properties (2) are boundary conditions.
Copula. Bivariate copula is a function 𝐶(𝑢, 𝑣) from 𝐼 2 to 𝐼, which is defined by the following conditions
2-increasing condition;𝐶(𝑢2 , 𝑣2 ) − 𝐶(𝑢2 , 𝑣1 ) − 𝐶(𝑢1 , 𝑣2 ) + 𝐶(𝑢1 , 𝑣1 ) ≥ 0,

(1)

for (𝑢𝑖 , 𝑣𝑗 ) ∈ 𝐼 2 (𝑖, 𝑗 = 1,2), 𝑢1 ≤ 𝑢2 , 𝑣1 ≤ 𝑣2 .
Boundary conditions;𝐶(𝑢, 0) = 𝐶(0, 𝑣) = 0,𝐶(𝑢, 1) = 𝑢 and 𝐶(1, 𝑣) = 𝑣.

(2)

The following Sklar’s theorem is the core theory among various copula theories. Thanks to this theorem we canconstruct multivariate distribution by coupling univariate marginal distributions.
Sklar’s theorem. Let H be a bivariate joint distribution function with marginal distribution function F and G; that islim 𝐻(𝑥, 𝑦) = 𝐹(𝑥) = 𝑢 and lim 𝐻(𝑥, 𝑦) = 𝐺(𝑦) = 𝑣
𝑦→∞

𝑥→∞

Then there exists a copula, which is uniquely determined on Ran F × Ran G such that
𝐻(𝑥, 𝑦) = 𝐶 (𝐹(𝑥), 𝐺(𝑦)) = 𝐶(𝑢, 𝑣).

(3)

Conversely, if C is a copula and F and G are distribution functions, then the function H defined by (3) is a bivariate joint
distribution function with marginal distribution functions F and G.
There are copulas for sample data, which are called empirical copulas. Empirical copulas are used for evolution of
copulas in discrete processes. We explain empirical copulas, empirical copula frequency functions and the relation
between them with reference to Nelson, R.B. [14].
Empirical copulas. Let {𝑥𝑘 , 𝑦𝑘 }𝑛𝑘=1 denote a sample with size n from a continuous bivariate distribution. The empiricalcopulas 𝐶𝑛 is given by
number of pairs (𝑥, 𝑦)in sample with 𝑥 ≤ 𝑥(𝑖) , 𝑦 ≤ 𝑦(𝑗)𝑖 𝑗𝐶𝑛 ( , ) =,𝑛 𝑛𝑛2
The empirical copula frequency function 𝑐𝑛 is given by
2

(4)

1
𝑖 𝑗
𝑐𝑛 ( , ) = { 𝑛 , 𝑖𝑓 (𝑥(𝑖) , 𝑦(𝑗) ) is an element of the sample,
𝑛 𝑛0, otherwise
where 𝑥(𝑖) and 𝑦(𝑗) , 1 ≤ 𝑖, 𝑗 ≤ 𝑛, denote order statistics from the sample.
Furthermore the relation between 𝐶𝑛 and 𝑐𝑛 is deduced by the definition as
𝑖
𝑗
𝑖 𝑗
𝑝 𝑞
𝐶𝑛 ( , ) = ∑∑𝑐𝑛 ( , ) , and
𝑛 𝑛
𝑛 𝑛
𝑝=1
𝑞=1

(5)

𝑖 𝑗
𝑖 𝑗
𝑖−1 𝑗
𝑖 𝑗−1
𝑖−1 𝑗−1𝑐𝑛 ( , ) = 𝐶𝑛 ( , ) − 𝐶𝑛 (, ) − 𝐶𝑛 ( ,) + 𝐶𝑛 (
,
).
𝑛 𝑛
𝑛 𝑛
𝑛 𝑛
𝑛 𝑛
𝑛
𝑛

2.2 Basic concept of rank correlations
Rank correlations are a sort of dependence measures for two variables, and their special feature is that they do notdepend on marginal distributions, but depend only on their copulas. That is to say, copulas determine rank correlations
We introduce Kendall’s tau (𝜏) and Spearman’s rho (ρ), which are typical rank correlations. Their definitions, their
popular version derived by copulas and their discrete versions are as follows.
Kendall’s tau. Kendall’s tau for a pair (𝑋, 𝑌) is the probability of concordance minus the probability of dis- concordance.
Kendall’s tau is defined as
𝜏𝑋,𝑌 ≔ 𝑃[(𝑋1 − 𝑋2 )(𝑌1 − 𝑌2 ) > 0] − 𝑃[(𝑋1 − 𝑋2 )(𝑌1 − 𝑌2 ) < 0],where (X1 , Y1 ) and (X2 , Y2 ) are independent and identically distributed random vector with joint distribution.
Moreover the popular version of Kendall’s tau is derived using only copula as
𝜏𝑋,𝑌 = 4 ∬ 𝐶(𝑢, 𝑣)𝑑𝐶(𝑢, 𝑣) − 1,

(6)

𝐼2

where 𝑢 and 𝑣 are uniform random variables, and their discrete version are derived by solely empirical copula as
𝑛
𝑛
𝑖−1
𝑗−1
2𝑛
𝑖 𝑗
𝑝 𝑞
𝑖 𝑞
𝑝 𝑗
∑ ∑ ∑
∑
[𝑐𝑛 ( , ) 𝑐𝑛 ( , ) − 𝑐𝑛 ( , ) 𝑐𝑛 ( , )]𝑛 − 1 𝑖=2 𝑗=2 𝑝=1 𝑞=1
𝑛 𝑛
𝑛 𝑛
𝑛 𝑛
𝑛 𝑛

=

(7)

𝑛
𝑛
2𝑛
𝑖 𝑗
𝑖−1 𝑗−1𝑖 𝑗−1
𝑖−1 𝑗∑ ∑ [𝐶𝑛 ( , ) 𝐶𝑛 (,) − 𝐶𝑛 ( ,) 𝐶𝑛 (, )].
𝑛 − 1 𝑖=2 𝑗=2
𝑛 𝑛
𝑛
𝑛
𝑛 𝑛
𝑛 𝑛

Spearman’s rho. Spearman’s rho for (𝑋1 , 𝑌1 ) and (𝑋2 , 𝑌3 ) is proportional to the probability of concordance minus theprobability of dis-cordance. Spearman’s rho is defined as
𝜌𝑋,𝑌 : = 3(𝑃[(𝑋1 − 𝑋2 )(𝑌1 − 𝑌3 ) > 0] − 𝑃[(𝑋1 − 𝑋2 )(𝑌1 − 𝑌3 ) < 0]),where (𝑋1 , 𝑌1 ), (𝑋2 , 𝑌2 ) and (𝑋3 , 𝑌3 ) are independent and identically distributed random vectors with a common joint
distribution functions.
The well-known version of Spearman’s rho is derived only by copula as
𝜌𝑋,𝑌 = 12 ∬ 𝑢𝑣 𝑑𝐶(𝑢, 𝑣) − 3 = 12 ∬ 𝐶(𝑢, 𝑣)𝑑𝑢𝑑𝑣 − 3,
𝐼2

𝐼2

(8)

where 𝑢 and 𝑣 are uniform random variables, where 𝑢 and 𝑣 are uniform random variables, and their discrete version
is derived by solely empirical copula as
12𝑛2 − 1

𝑛

∑

𝑖=1

𝑖 𝑗
𝑖 𝑗
[𝐶𝑛 ( , ) − ∙ ] .
𝑛
𝑛
𝑛
𝑛
𝑗=1
𝑛

∑

3

(9)

3. Evolution of copulas
In this section, we summarize continuous evolution of bivariate copulas and discrete evolution of bivariate copulas.
The reason we adopt the heat equation to evolve copulas is as follows.
a) Rank correlations, one of the prevailing measures of dependence, are derived only by copulas. That is to say,
copulas determine rank correlations. Thus it is natural to analyze only copulas in the study of transformations ofdependence structures through time.
b) The object of our study is autonomous movements of dependencies. The heat equation is one of the basic
partial differential equations used to describe dynamic autonomous movements. Therefore we adopt it as a first step.
3.1 Evolution of copulas in continuous processes
We proved the existences and solutions of evolution of copulas in continuous processes, and their convergence tothe product copula. Moreover we proved that rank correlations of evolution of copulas converge to zero exponentially as
𝑡 → ∞. See Yoshizawa,Y.[23], Yoshizawa, Y., & Ishimura, N. [21] and Ishimura, N., & Yoshizawa, Y. [7],[8].
Continuous evolution of copulas. For any bivariate copula 𝐶0 (𝑢, 𝑣), there exists a unique family of time dependent
bivariate copula {𝐶(𝑢, 𝑣, 𝑡)}𝑡≥0 , which satisfies the heat equation∂𝐶𝜕2
𝜕2(𝑢, 𝑣, 𝑡) = ( 2 + 2 ) 𝑐(𝑢, 𝑣, 𝑡),∂𝑡𝜕𝑥𝜕𝑦for (𝑢, 𝑣, 𝑡) ∈ 𝐼 2 × (0, ∞), 𝑤ℎ𝑒𝑟𝑒 𝐶(𝑢, 𝑣, 𝑡) = 𝐶0 (𝑢, 𝑣) 𝑜𝑛 (𝑢, 𝑣) ∈ 𝐼 2 .
The solution of the partial differential equation (10) is

(10)

∞
2

2

2

𝐶(𝑢, 𝑣, 𝑡) = 𝑢𝑣 + 4 ∑ 𝑒 −π (𝑚 +𝑛 )𝑡 sin 𝑚π𝑢 sin 𝑛π𝑣 𝐾(𝑚, 𝑛) ,

(11)

𝑚,𝑛=1

where 𝐾(𝑚, 𝑛) = ∬𝐼2 sin 𝑚π𝜉 sin 𝑛π𝜂 (𝐶0 (𝜉, 𝜂) − 𝜉𝜂)𝑑𝜉 𝑑𝜂 .
Convergence to the product copula. Evolution of bivariate copulas converge to the product copula, which means
independences between variables, as
uniformly on (𝑢, 𝑣) ∈ 𝐼 2 ,lim 𝐶(𝑢, 𝑣, 𝑡) = 𝛱(𝑢, 𝑣)𝑡→∞where 𝛱(𝑢, 𝑣) ∶= 𝑢𝑣.

(12)

Rank correlation. Rank correlations are a sort of dependence measures, and Kendall’s tau and Spearman’s rho are typicalrank correlations. Their special feature is that they do not depend on marginal distributions, but depend only on their
copulas. Furthermore rank correlations, Kendall’s tau (𝜏𝐶𝑡 ) and Spearman’ rho (ρ𝐶𝑡 ), of evolution of copulas 𝐶(𝑢, 𝑣, 𝑡),converge to zero exponentially as 𝑡 → ∞.

3.2 Evolution of copulas in discrete processes
In general, it is difficult to solve partial differential equations analytically; therefore, numerical approaches are often
applied in practice, especially where analytical solutions do not exist. Moreover, numerical analysis is suitable forcomputer calculation.
Therefore we studied and created the following discrete evolution of bivariate copulas which satisfy the following
discrete version of the heat equation. See Yoshizawa,Y.[23] and Yoshizawa, Y., & Ishimura, N.[22].
Let 𝑁 ≥ 0 and 0 < ℎ ≤ 1, we put
𝑖

𝑗

𝑁

𝑁

△ 𝑢 =△ 𝑣 ∶=1
𝑁

= 𝑀, △ 𝑡 ∶= ℎ, 𝜆 ∶=

△𝑡
(△𝑢)2

=

△𝑡
(△𝑣)2

= ℎ𝑁 2 , and𝑢𝑖 ∶= 𝑖 △ 𝑢 = , 𝑣𝑗 ∶= 𝑗 △ 𝑣 = , for 𝑖, 𝑗 = 0,1, ⋯ , 𝑁.
𝑛
At any {(𝑢𝑖 , 𝑣𝑗 )}𝑖,𝑗=0,1,2,⋯,𝑁 , the value 𝐶𝑖,𝑗∶= 𝐶 𝑛 (𝑢𝑖 , 𝑣𝑖 ) is governed by the system of the difference equation
𝑛+1
𝑛
𝑛
𝑛
𝑛
𝑛
𝑛
𝑛
𝐶𝑖,𝑗
− 𝐶𝑖,𝑗𝐶𝑖+1,𝑗− 2𝐶𝑖,𝑗+ 𝐶𝑖−1,𝑗𝐶𝑖,𝑗+1− 2𝐶𝑖,𝑗+ 𝐶𝑖,𝑗−1=+,(△ 𝑢)2(△ 𝑣)2△𝑡

4

for 𝑖, 𝑗 = 0,1, ⋯ , 𝑁 − 1,0𝑛𝑛where 𝐶𝑖,𝑗
= 𝐶 0 (𝑢𝑖 , 𝑣𝑗 ) ∶= 𝐶0 (𝑢𝑖 , 𝑣𝑗 ) is an initial copula, together with the boundary conditions as 𝐶𝑖,0
= 𝐶0,𝑗=𝑛𝑛0, 𝐶𝑖,𝑁= 𝑢𝑖 , 𝐶𝑁,𝑗
= 𝑣𝑗 , for 𝑖, 𝑗 = 0,1, ⋯ , 𝑁,1

and the constrain 0 ≤ λ ≤ . The above deference equation is rewritten
4

as
𝑛+1
𝑛
𝑛
𝑛
𝑛
𝑛
𝐶𝑖,𝑗
= (1 − 4𝜆)𝐶𝑖,𝑗+ 𝜆(𝐶𝑖+1,𝑗+ 𝐶𝑖−1,𝑗+ 𝐶𝑖,𝑗+1+ 𝐶𝑖,𝑗−1)(13)

The image of the deference equation (13) is shown in Figure 1.
Furthermore we define the interpolation as𝐶𝑛𝑛
𝐶 𝑛 (𝑢, 𝑣) ∶= 𝐶𝑖,𝑗
+ 𝑖+1,𝑗𝑛−𝐶𝑖,𝑗𝑢𝑖+1 −𝑢𝑖(𝑢 − 𝑢𝑖 ) +𝑛𝑛
𝐶𝑖,𝑗+1−𝐶𝑖,𝑗𝑣𝑗+1 −𝑣𝑗(𝑣 − 𝑣𝑗 ) +𝑛𝑛𝑛𝑛
𝐶𝑖+1,𝑗+1−𝐶𝑖+1,𝑗−𝐶𝑖,𝑗+1+𝐶𝑖,𝑗(𝑢𝑖+1 −𝑢𝑖 )(𝑣𝑗+1 −𝑣𝑗 )(𝑢 − 𝑢𝑖 )(𝑣 − 𝑣𝑗 ),(14)

for 𝑢𝑖 ≤ 𝑢 ≤ 𝑢𝑖+1 , 𝑣𝑗 ≤ 𝑣 ≤ 𝑣𝑗+1 and (𝑢, 𝑣) ∈ 𝐼 2 .
𝐶𝑖𝑗𝑛+1
𝑛
𝐶𝑖𝑗+1

𝑛
𝐶𝑖−1𝑗

𝐶𝑖𝑗𝑛
𝑛
𝐶𝑖+1𝑗
𝑛
𝐶𝑖𝑗−1

Figure 1. Imagae of deference equation

Discrete evolution of bivariate copulas. For any initial copula 𝐶0 , there exists a sequence of copula{𝐶 𝑛 (𝑢, 𝑣)}𝑛=0,1,2,⋯

on (𝑢, 𝑣) ∈ 𝐼 2 ,which satisfy the system of difference equation at every {(𝑢𝑖 , 𝑣𝑗 )}𝑖,𝑗=0,1,2,⋯,𝑁 andthe above interpolation. We can call these copulas 𝐶 𝑛 (𝑢, 𝑣) as evolution of bivariate copulas in discrete processes, which
satisfy 2-increasing condition (1) and boundary conditions (2).
Convergence to the product copula. We proved that discrete evolution of copulas 𝐶 𝑛 (𝑢, 𝑣) converge to the productcopula 𝛱(𝑢, 𝑣) = 𝑢𝑣 uniformly on 𝐼 2 as well as continuous evolution of copulas. It is remarkable that the convergences
do not depend on the fineness of mesh 𝑀, but depend only on the number of times 𝑛.
Rank correlation. For any initial copulas 𝐶0 and a sequence of discrete evolution of copulas {𝐶 𝑛 (𝑢, 𝑣)}𝑛=1,2,⋯,
Kendall’s tau (𝜏𝑛 ) and Spearman’ rho (ρ𝑛 ) converge to zero exponentially as n → ∞, corresponding to the continuous
type.

Convergence from discrete to continuous. We proved that discrete evolution copulas converge to continuous evolutionof copulas as 𝑁 → ∞ and ℎ → 0 uniformly on (𝑢, 𝑣) × 𝑡 ∈ 𝐼 2 × (0, ∞). Thus we can treat the discrete evolution of
copulas as an approximation of the continuous evolution of copulas.

3.3 Application to empirical data
The evolution of copulas has properties to suit events whose dependence monotonically moving. Therefore, we focus
on rapidly changing events when their directivities are almost stable. We select foreign exchange rates on January 15, 2015,
when the Swiss franc endured a shock breakout after the announcement that the Swiss central bank had stopped monetary5policy efforts to maintain the Swiss franc against the euro at more than 1.20. Therefore we analyze the dependence of euro–
Japanese yen foreign exchange rates with those of the Swiss franc–Japanese yen.
First we collect data on the second time scale in order to capture their monotonic directivity. We construct empiricalcopulas of the euro–Japanese yen rates and the Swiss franc–Japanese yen rates for every second of 40 minutes using theformula (4). Then, we calculate their Kendall’s tau correlation measure by using the formula (7). We apply a smoothing
technique, a moving average method, to the transitions of Kendall’s tau, since they fluctuate and include some singular
data.
Next we compare Kendall’s tau of the evolution of empirical copulas to the abovementioned moving averages of
Kendall’s tau of empirical copulas. We choose the start time when Kendall’s tau of the empirical copula and its moving
average are almost equal. We evolve the empirical copula at the start time using the deference equation (13) for 20 minutes.
For reference, we plot the results again in Figure 2, which shows that the evolution of empirical copulas approximate thesmoothed transition of empirical copulas from the viewpoint of Kendall’s tau. See Ishimura, N., & Yoshizawa, Y. [11] andYoshizawa,Y. [23] for the details.
Evolution of Tau

0.8

Tau Ave600

0.6
0.4
0.2
0
Notes: TauAve600: Moving averages of 600 datasets of Kendall’s tau for every second.
Evolution of Tau: Kendall’s tau of evolution of copulas. We extract the empirical copula at around 18:30 when the Kendall’s tau of theempirical copulas is almost equal to the smoothed Kendall’s tau. We evolve them 1,200 times, which means for 20 minutes. For this1
evolution, we use the deference equation (13) with 𝜆= .
5Data source: Bloomberg, exchange rate.

Figure 2. Kendall’s tau of evolution of copulas

Using these flexible discrete copula models, we analyze the movement of dependence relations between the Swiss
franc and the euro against the Japanese yen approximately. We can verify the practicality of some theories of the evolutionof copulas, although this is restricted to events whose essential dependence does not fluctuate but transforms
monotonically.

4. Generalized evolution of bivariate copulas in discrete processes
We are then able to generalize the difference equation (13) so that the asymmetrical weight is allowed as
𝑛+1
𝑛
𝑛
𝑛
𝑛
𝑛
𝐶𝑖,𝑗
= 𝛼𝐶𝑖,𝑗+ (𝛽1 𝐶𝑖+1,𝑗+ 𝛽2 𝐶𝑖−1,𝑗+ 𝛽3 𝐶𝑖,𝑗+1+ 𝛽4 𝐶𝑖,𝑗−1),4where 𝛼 > 0, 𝛽𝑘 > 0 (𝑘 = 1,2,3,4), and 𝛼 + ∑𝑘=1 𝛽𝑘 = 1.
As for the detail, see Ishimura, N., & Yoshizawa, Y. [9].

(15)Moreover we generalize this concept to propose the recurrence relation as
𝐿

𝑛+1
𝐶𝑖,𝑗
:= ∑

𝑘=−𝐿

𝐿

∑

𝑙=−𝐿

𝑛
𝑛
𝛼𝑘,𝑙
𝐶𝑖+𝑘,𝑗+𝑙,(16)

for any n,𝑛𝑛𝛼𝑘,𝑙≥ 0 and ∑𝐿𝑘=−𝐿 ∑𝐿𝑙=−𝐿 𝛼𝑘,𝑙
= 1 for any L ≤ N, and𝑛𝑛𝑛𝐶𝑖,𝑗 = 0 for 𝑖 ≤ 0 𝑜𝑟 𝑗 ≤ 0, and 𝐶𝑖,𝑗= 𝑢𝑖 for 𝑗 ≥ 𝑁 and 𝐶𝑖,𝑗= 𝑣𝑗 for 𝑖 ≥ 𝑁.
First, we propose the recurrence relation (16) succeeds the conditions of copulas, and prove it in the following
proposition 1.
6

𝑛
𝑛+1
Proposition 1. If 𝐶𝑖,𝑗satisfies the conditions of copulas (1) and (2), then 𝐶𝑖,𝑗defined by the recurrence relation (16)also satisfies the conditions of copulas.
𝑛𝑛𝑛+1(Proof) 𝐶𝑖+𝑘,𝑗+𝑙
satisfies 2-increasing condition and 𝛼𝑘,𝑙≥ 0 , thus 𝐶𝑖,𝑗also satisfies 2-incresing condition.
𝑛+1
𝑛+1
𝐶𝑖,𝑗 satisfies the boundary conditions according to its definition (16). Hence 𝐶𝑖,𝑗satisfies the conditions of copulas (1)and (2).

∎
Secondly, we introduce the generalized evolution of bivariate copulas in discrete processes, generated by therecurrence relation and the interpolation , and prove that they are copulas in the theorem 2.
Theorem 2. (Generalized evolution of bivariate copulas in discrete processes). For any initial copula 𝐶0, there exists a
sequence of copulas {𝐶 𝑛 (𝑢, 𝑣)}𝑛=0,1,2,⋯

on (𝑢, 𝑣) ∈ 𝐼 2 ,which satisfy the system of the recurrence relation (16) at every{(𝑢𝑖 , 𝑣𝑗 )}𝑖,𝑗=0,1,2,⋯,𝑁 and the interpolation (16). We can call these copulas 𝐶 𝑛 (𝑢, 𝑣) as the generalized evolution ofbivariate copulas in discrete processes, which satisfy 2-increasing condition (1) and boundary conditions (2).
𝑛𝑛(Proof) 𝐶(𝑢, 𝑣) is a function from 𝐼 2 to 𝐼, because 𝛼𝑘,𝑙
≥ 0 and ∑𝐿𝑘=−𝐿 ∑𝐿𝑙=−𝐿 𝛼𝑘,𝑙
= 1 for any L ≤ N.
𝑛+1
𝐶𝑖,𝑗 satisfies the boundary conditions by the proposition 1, thus 𝐶(𝑢, 𝑣) also satisfies the boundary condition.
We verify 2-increasing condition. Let 𝑢𝑖 ≤ 𝑢1 ≤ 𝑢2 ≤ 𝑢𝑖+1 , 𝑣𝑗 ≤ 𝑣1 ≤ 𝑣2 ≤ 𝑣𝑗+1 , then using the proposition 1(𝑢 −𝑢1 )(𝑣2 −𝑣1 )≥ 0.
𝑖+1 −𝑢𝑖 )(𝑣𝑗+1 −𝑣𝑗 )𝑛𝑛𝑛𝑛
𝐶 𝑛 (𝑢2 , 𝑣2 ) − 𝐶 𝑛 (𝑢2 , 𝑣1 ) − 𝐶 𝑛 (𝑢1 , 𝑣2 ) + 𝐶 𝑛 (𝑢1 , 𝑣1 ) = (𝐶𝑖+1,𝑗+1− 𝐶𝑖+1,𝑗− 𝐶𝑖,𝑗+1+ 𝐶𝑖,𝑗) (𝑢 2Hence we prove that {𝐶 𝑛 (𝑢, 𝑣)}𝑛=0,1,2,⋯ are copulas.

∎
Finally, as an example we show a generalized evolution of bivariate copula in discrete processes, which strengthen
dependence relations through time, in the following example 3. For simplicity, we omit the interpolation (14).
𝑛𝑛𝑛Example 3. Let N=100, L=2, we set 𝛼𝑘,𝑙as 𝛼2,2= 1 , 𝛼𝑘,𝑙
= 0 for k≠ 2 and l ≠ 2. According to 2-increasing𝑛properties of 𝐶𝑖,𝑗 , the following equation holds as
𝑛+1
𝑛
𝐶𝑖,𝑗
= max−𝐿≤𝑘≤𝐿,−𝑙≤𝑙≤𝐿 𝐶𝑖+𝑘,𝑗+𝑙.
Thus we confirm that this generalized evolution of copula increase their dependences as time passing.

5. Discussion
In this paper, we extend our previous work, the evolution of bivariate copulas, to the generalized evolution of
copulas in discrete processes. The traditional evolution of bivariate copulas have the properties to decrease dependence
monotonically over time, therefore they fit with events whose dependence decrease. Applying the above generalized
evolution of copula, we can manage the events whose dependences fluctuate. According to the mathematical equation(16) with the ideal coefficient 𝛼𝑛𝑘,𝑙 , we can construct the dependence structure model transforming autonomously throughtime. The difficulty is to assume the proper coefficients 𝛼𝑛𝑘,𝑙 for any 𝑛, 𝑘, 𝑙.
Recently theory, technique and computer power for Artificial Intelligence (AI) are increasing their capacities, andAI becomes to be used widely. We think that AI, such as machine learning or deep learning, will help to solve thedifficult problem. We hope that collaboration between the generalized evolution of copulas and AI will contribute to thestudy of various kinds of dependence structures transforming autonomously.

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