On copulas of self-similar Ito processes


 We characterize the cumulative distribution functions and copulas of two-dimensional self-similar Ito processes, with randomly correlated Wiener margins, as solutions of certain elliptic partial differential equations.


Introduction
The paper deals with the borderline of the copula theory and stochastic processes. It concerns the vector valued stochastic processes X t = (X t , . . . , X n t ), t ∈ T. The goal is to describe the evolution of interdependences between X t , . . . , X n t in terms of copulas and copula processes. In a way it is a continuation of papers by Sempi [31], Choe etal. [7], Jaworski and Krzywda [16] and Jaworski [15]. Sempi in ( [31]) was studying the possibility of coupling two Wiener processes by using a given copula. In [16] we started to investigate copulas of self-similar processes. In [7] the partial di erential equation for copulas of the Ito processes is derived but under very tight technical assumptions. In [15] it is achieved but in a general case. It is shown that the copula process is a weak solution of a parabolic partial di erential equation.
In this paper we study the evolution of the dependence between two Wiener processes with random quadratic covariation. Speci cally we extend the research on two-dimensional 1/2-self-similar Ito di usions whose margins are Wiener processes and their interdependencies at every time moment are described by a given copula, that we have initiated in [16]. We drop the assumption of di erentiability of cumulative distribution functions and corresponding copulas. We replace the classical derivatives by weak derivatives in Sobolev sense, so called distributional derivatives, as in [15]. This allows us to present new and more general results. We consider 1/2-self-similar processes as solutions of certain stochastic di erential equations (SDE) and show how to construct 1/2-self-similar solution basing on a non-self-similar solution.
In more details, we consider a pair of stochastic processes (X t , Y t ) t≥ , where X = (X t ) t≥ and Y = (Y t ) t≥ are one dimensional Wiener processes and their quadratic covariation is homogeneous for where the real valued function A is de ned on the real plane and |A| is a bounded by 1. Such processes arise as solutions of the stochastic di erential equations (SDE) with homogeneous coe cients where B(x, y) = − A(x, y) .
We begin with showing that for A and B Lipschitz, any pair of Wiener processes ful lling 1.2 gives rise to / -self similar solutions. They appear as limits of time-rescaled (time-shifted) processes (X tτ / √ τ, Y tτ / √ τ) t≥ when τ tends to in nity or as weighted generalized mixtures of time-rescaled processes. Then we derive a partial di erential equation (PDE) (satis ed in a weak sense) that describes the copula of the initial value, i.e. the copula C of the random pair (X , Y ), giving rise to / -self similar solution.
where Φ and φ denote the cumulative distribution function and the density of the standard normal distribution N( , ) and D i,j denote the partial derivatives. We also present the opposite result by which if any copula C ful lls the previously obtained PDE in a weak sense with coe cients satisfying certain regularity conditions then there exists a / -self-similar process (X t , Y t ) t≥ such that the copula of (X t , Y t ) t≥ is equal to C. Furthermore basing on the maximum principle for the elliptic PDE we show that the point-wise dominance of the quadratic covariations A implies the concordance dominance of the corresponding copulas. The structure of the paper is as follows: On the start we shortly recall basics on copulas, self-similarity and weak derivatives (section 2). In section 3 we introduce the underlying stochastic processes X t and Y t and state the results concerning their self-similarity. First concerning the existence of 1/2-self-similar solutions among solutions of certain stochastic di erential equations, next concerning their di erential characterization. The proofs are provided in section 4. In the last section we discuss some examples based on Gaussian, FGM and Archimedean copulas.

Notation
We recall the basic concepts.
A (bivariate) copula is a restriction to [ , ] of a distribution function whose univariate margins are uniformly distributed on [ , ]. Speci cally, C : [ , ] → [ , ] is a copula if it satis es the following properties: Due to the celebrated Sklar's Theorem, the joint distribution function F of any pair (X, Y) of random variables de ned on the probability space (Ω, F, P) can be written as a composition of a copula C and the univariate marginals F and F , i.e. for all (x, y) ∈ R , F(x, y) = C(F (x), F (y)). Moreover, if X, Y are random variables with continuous cumulative distribution functions, then the copula C is uniquely determined. We will denote the set of all two-dimensional copulas by C . Note, that C is a bounded, compact, convex subset of the Banach space of the continuous functions on the unit square endowed with the supremum metric. For copulas C and C we put (2.1) Speci cally any sequence of copulas contains a convergent subsequence, i.e.
The copulas C * , being the limits of subsequences, are referred to as cluster points of the sequence Cn. For more details about copula theory and some of its applications, we refer to [6,9,[17][18][19][20]26].
A random process is self-similar if its distributions scale. Speci cally where ∼ denotes equality of joint distributions. We call H the exponent of self-similarity of the process. For example standard Brownian Motion is / -self-similar. For the details on self-similarity we refer to [10] and [33]. Please note that our de nition is a bit weaker than the common literature de nition, ie. we do not require the equality of distributions of stochastic processes but only those of random vectors at each time t ≥ t . We can re-write the de nition of self-similarity as follows where F t is a cumulative distribution function of the pair (X t , Y t ) and F = F t . By Sklar's theorem applied to where we use Φ = Φ t and Ψ = Ψ t to denote univariate cumulative distribution functions of X t and Y t which we assume to be continuous. Since the processes X t and Y t also are self-similar analogously we may Once again by Sklar's theorem for each t ≥ t we obtain a copula C t such that Therefore, by the uniqueness of copula for random variables with continuous distribution functions, we have We conclude that, when X t and Y t have continuous distribution functions for each t ≥ t , then the process (X t , Y t ) t≥t is self-similar if and only if its copula is constant and both (X t ) t≥t and (Y t ) t≥t are self-similar.
In the following we shall deal with H = .
To characterize the copulas of 1/2-self-similar processes we will need to substantially weaken the notion of partial derivatives (see [4,11,13] We say that v is α th weak partial derivative of u, written for all test functions h ∈ C ∞ (U) with compact support and |α| = α + · · · + αn.
Weak partial derivatives, if they exist, are unique up to a set of measure zero. Note that, since copulas are Lipschitz functions, they are weakly di erentiable. As an example of weak partial derivatives of a copula C(u, v) one may consider one of the Dini derivatives ( [8,24]), say right-side upper one, applied respectively to both variables The choice of the version of the weak derivative is related to the choice of the version of the conditional probability. In more details, if C is a copula of the random variables X and Y, then the formula expressing the conditional distribution function of X in terms of the right continuous modi cation of the weak derivative (2.4), determines the version of the conditional probability P(·|σ(Y)) on the σ-eld σ(X) (compare [2,14]). Since the modi cations occur only at "jumps" of D , C(F X (·), F Y (y)), we have for xed y, such that

Main results
We consider the solutions (X t , Y t ) t≥ of the following system of stochastic di erential equations: where A1. W and W are two independent Wiener processes de ned on the probability space (Ω, F, P), A2. the functions A, B : R → R are Lipschitz, A3. for all (x, y) ∈ R we have B(x, y) = − A(x, y) and |A(x, y)| < , A4. A is di erentiable with respect to the second variable.
By M A and L A we shall denote respectively the supremum of the modulus and the Lipschitz coe cient of A with respect to the euclidean distance Furthermore we denote by L the square of the Lipschitz coe cient of the vector (A, B) Note that when M A < , we have a bound In fact we understand (3.1) as an integral equation: where the initial values, i.e. the random pair (X , Y ), and the two-dimensional Wiener process (W t − W , W t − W ) t≥ are independent. The process (X t , Y t ) t≥ is adapted to the ltration de ned by the Wiener processes W and W and the initial values Assumption A1. ensures that the Ito integrals in (3.6) are well de ned and the quadratic covariation of any solution of SDE (3.1) The second one, A2., implies the existence and uniqueness of the solution of SDE (3.1). Due to A3., the quadratic variation of Y = (Y t ) t≥ is given by which implies that (Y t+ − Y ) t≥ is a Wiener process. Our goal is to study the existence and properties of / -self-similar solutions of SDE (3.1). Note that the self-similarity of the process (X t , Y t ) t≥ does not depend on the choice of the probability space (Ω, F, P) neither on the choice of the Wiener processes. It depends only on the joint distribution of the of the pair (X , Y ). It is a corollary from the uniqueness of the weak solutions of the SDE, see [5]. In more details, when the seven tuple is a weak solution of equations (3.1) and Please also take note of the fact that the margins (X t ) t≥ and (Y t ) t≥ are / -self-similar if and only if they coincide with one-dimensional Wiener processes. The proof is elementary, we refer to section 4.1. Throughout the rest of this paper we denote by Φ and φ the cumulative distribution function and density of the standard normal distribution N( , ).
On the other hand, as it is shown in section 4.3, the set of / -self-similar solutions is not empty. Indeed for any functions A and B ful lling A2 and A3 there exits a selfsimilar solution. Such / -self-similar solutions arise as a weighted generalized mixtures of τ-rescaled processes When the functions A and B = √ − A , i.e. the volatility of the solutions of (3.1), are varying in a moderate way then the probability law of "properly chosen" initial values can be described just as a limit with respect to convergence in distribution. Let C C t denote a copula of the pair (X t , Y t ) for t ≥ , where the process (X t , Y t ) t≥ is a solution of eq. (3.1), with initial values X and Y having the standard normal distribution X ∼ Y ∼ N( , ) and linked by a copula C.

Proposition 3.3. If L, the square of the Lipschitz coe cient of (A, B), is smaller than 1, then for any copula C a solution of eq. (3.1), with initial values having a cumulative bivariate distribution C
The proof is provided in section 4.3. Note that L < when for example L A + A M < . Under assumption L < , Proposition 3.3 implies, that the rescalings (shiftings) of any solution of eq. (3.1) are converging in distribution to a self similar solution. This has a certain practical impact. When we observe some phenomena driven by a solution of SDE (3.1), with L < , for a su ciently long time we may approximate the "true" solution by a 1/2-self-similar one. In the last part of the section 5.1 we provide an illustration of this e ect.
After this short discussion on existence we state our results on di erential characterization of copulas of / -selfsimilar solutions.
Theorem 3.4. Assume A1,A2, A3 and A4, then: If the process (X t , Y t ) t≥ is a / -self-similar solution of eq. (3.1), then the copula of (X , Y ) which we denote by C(u, v) is twice di erentiable in a weak sense in ( , ) and almost everywhere in ( , ) ful lls the equation (3.11) the cumulative distribution function of (X , Y ) which we denote by F(x, y) is twice di erentiable in a weak sense and ful lls the equation As a consequence of the maximum principle for elliptic PDE, we get:

If copulas C and C ful ll PDE (3.11) with A equals respectively to A and A and
then copula C dominates in concordance ordering  Corollary 3.7. Let a twice weakly di erentiable copula C be a solution of PDE (3.11) with coe cient A and Gaρ, ρ ∈ (− , ), be a Gaussian copula with correlation coe cient ρ.
The "existence" Theorem 3.2 and the "uniqueness" Corollary 3.6 imply the following theorem.
The "uniqueness" Corollary 3.6 allows us to restate Theorem 3.2 in a more e ective way.
The proofs of the subsequent theorems and corollaries stated above are provided in sections 4.5, and 4.6.
With the exception of Corollary 3.7 which is based on Section 5.1 The weak derivatives proved also to be very useful in a study concerning dynamics of copulas of more general Ito processes, see [15].
In section 5, we will show that the set of solutions of equations (3.11) contains Gaussian copulas with ρ < , FGM copulas with α < and some but not all Frank copulas and does not contain Clayton copulas. In the rst case A is constant, it equals to correlation coe cient. In the second and third A may vary.

Proofs and auxiliary results
. Margins -proof of Proposition 3.1 , then X t , for t > , has normal distribution, with zero mean, as well. Furthermore Hence (X t ) t≥ is / -self-similar, ie,: X t ∼ √ tX . Now let (X t ) t≥ be / -self-similar. We need to show that X ∼ N( , ). We will base on the properties of characteristic functions (see [3] §26). Since we get for xed s Since X √ t converges to 0 when t → +∞, we obtain by Lebesgue dominated convergence Theorem Hence X has standard normal distribution.
When it comes to the second variable, the stochastic process we can see that by repeating the above reasoning we may conclude that Y t is / -self-similar if and only Y ∼ N( , ).

. Semigroup property
with initial values X and Y having the standard normal distribution X ∼ Y ∼ N( , ) and linked by a copula C. Basing on Proposition 4.2 we show that the mapping is a representation of a multiplicative semigroup [ , ∞), ·, (see section 4.3 for more details). Observe that copula of a 1/2-self-similar process is a xed point of H. It shows that such copulas can be obtained as generalized weighted averages along the orbit of H.
We start with the following auxiliary proposition which will be essential in the course of proving Theorem 3.2. We show that the rescaling (shifting) of a strong solution of SDE (3.1) give rise to a weak solution of the same equations. Proof.
First we observe that for τ > Next by the change of variables (s := uτ) we get Since for xed τW are independent Wiener processes (de ned on the same probability space as W and W ), we conclude that indeed we obtained a solution of the stochastic di erential equations (3.1) with another pair of Wiener processes with initial values (τ − / Xτ , τ − / Yτ).
Due to the uniqueness of the weak solutions of the SDE (see [5]) Proposition 4.1 implies Proposition 4.2.

Proposition 4.2.
We assume A1, A2 and A3. Then the solution is law equivalent to the τ-rescaled process

. Existence
We assume A1, A2 and A3 throughout this section.
First we establish some estimates which imply the continuity of solutions of equations (3.1) in L norm.
In more details, since both (X t ) t≥ and (Y t ) t≥ coincide with Wiener processes we have for s, t ≥ Let us consider two processes (X t , Y t ) t≥ and (X t , Y t ) t≥ which solve equation (3.1) with di erent initial values: 3) where B(x, y) = − A(x, y) . By the following lemma the distances between (X t ) t≥ and (X t ) t≥ and also between (Y t ) t≥ and (Y t ) t≥ are linear functions of the distances between the initial values.

Lemma 4.3. For any T ≥ :
where L is a sum of squares of Lipschitz coe cients for A and B, Proof.
Equality (4.7) follows readily since For the proof of (4.8), we apply subsequently the Ito formula, Lipschitz inequality and (4.9). Finally we The Gronwall Lemma (see [11] Appendix B.j) applied for a function Having substituted t = T − we conclude the proof of Lemma.
To prove the existence of self-similar solutions, we follow the approach of Khasminskii ( §2.2 [21]) and construct a self-similar solution basing on the averages of a process.
In more details, let C be any copula. We shall consider the solutions (X t , Y t ) t≥ of the set of equations (3.1) with initial values (X , Y ), for which we assume that both X and Y are standard normal N( , ) and the copula of their joint distribution is C. By H(t, C) we denote the copula of (X t , Y t ). Since X t and Y t have normal distribution with mean 0 and variance equal t (N( , t)), the joint cumulative distribution of the pair (X t , Y t ) is given by (4.14) Since for any copula C there exists a pair of random variables with joint distribution function C(Φ(x), Φ(y)), the function H is well de ned on [ , ∞) × C (compare [21] Th. 3.4).
For the solution (X t , Y t ) t≥ of (3.1) to be a / -self-similar process we need to show that H(t, C) = C for all t ≥ , therefore we are in fact interested in the existence of xed points of H.
In the following proposition we list the basic properties of H. Let {Cn} n∈N+ be any sequence of copulas convergent to copula C∞. Then, by the Skorohod Theorem, there exists a probability space (Ω , F , P ), a sequence of random pairs {(Z n , Z n )} n∈N+ and a random vector (Z ∞ , Z ∞ ) (all de ned on (Ω , F , P )), such that: 1. Cn(Φ(x), Φ(y)) is a distribution function of (Z n , Z n ), n = , , . . . , ∞; 2. (Z n , Z n ) almost surely converges to (Z ∞ , Z ∞ ).

H is continuous with
Let us consider the following product of probability spaces (Ω , F , P ) = (Ω × Ω , F × F , P × P ).
Since Wiener processes W , W have been de ned on (Ω, F, P) and the initial values Z i n on (Ω , F , P ) we extend them onto the product space by putting Next we analyze the stochastic equations (3.1) on the previously de ned product space, denoting by (X n,t , Y n,t ) their solutions with initial values (Z n , Z n ), for n = , , . . . , ∞.
Obviously H(t, Cn) are copulas of (X n,t , Y n,t ).
Note that due to Lebesgue dominated convergence Theorem the sequence {(Z n , Z n )} n∈N+ converges in L norm Therefore, by Lemma 4.3 and equalities (4.2), for any convergent sequence of indices {sn} n∈N+ ⊂ [ , ∞) the sequence of random pairs {(Xn,s n , Yn,s n )} n∈N+ converges in L to (X∞,s ∞ , Y∞,s ∞ ), where s∞ denotes the limit of sn.
By the following lemma, convergence in L and convergence in distribution are closely related. To simplify the notation we put for two random pairs (4.15)

Lemma 4.5. Let V and V be two random pairs with standard normal margins and copulas respectively C and C , de ned on the same probability space. Then
Proof.
Let (x, y) be any point of the real plane and ε a positive constant. Applying the elementary set theory and Markov inequality (see [3] formula (21.12)) we get the following estimate Since copulas are Lipschitz functions (with Lipschitz constant 1) and Φ is also Lipschitz function (with Lipschitz constant φ( ) = √ π ) we further estimate Since the bound is valid for all positive ε, we substitute which minimizes the estimate.
The above bound remains valid when we replace C and C . Moreover it is valid for all points (x, y) ∈ R . Therefore Thus convergence in L implies convergence in distribution and the joint cumulative distribution functions of (Xn,s n , Yn,s n ) are converging to the cumulative distribution functions of (X∞,s ∞ , Y∞,s ∞ ) We recall that a random pair Zµ is a mixture of random pairs Z θ , θ ∈ Θ, with respect to the probabilistic measure µ on Θ, when for every bounded Borel function f on R (4.21) We assume that the random pairs Zµ and Z θ , θ ∈ Θ, and the two-dimensional Wiener process (W t − W , W t − W ) t≥ are independent. We denote by (X z To conclude the proof of point 3 it is enough to select where Cµ(x, y) = Θ C θ (x, y)dµ(θ).
Next we show that the set of xed points of the semigroup H is not empty. We select a copula C and as a xed point take the generalized average of the trajectory of C. In more details. Proof. Due to Ascoli theorem there exists a subsequence of C k having a limit, i.e. a cluster point of the sequence. We denote this limit copula by C * C kn −→ C * .
Since H is continuous and commutes with the mixtures we get Therefore, since copulas are bounded by 1, we get for t ≥ The following corollary concludes the proof of Theorem 3.2.

Corollary 4.7. Let C * be a xed point of H. Then the solution of equations (3.1) with initial values
where Ψ is a generalized inverse of the weak derivative of C * Proof. Let F t (x, y) be a cumulative distribution function of the solution (X t , Y t ). Since we get from Proposition 4.4

. Proof of Proposition 3.3
In this section we assume that the sum of squares of Lipschitz coe cients of A and B, denoted by L, is smaller than 1. Let C * be a xed point of the semigroup H from Proposition 4.6 and (X t , Y t ) t≥ be a 1/2-self-similar solution of equations (3.1) from Corollary 4.7. Its distribution functions are equal to We compare it with an arbitrary solution of equations (3.1) (X t , Y t ) t≥ . Let be its distribution functions. Due to Lemmas 4.3 and 4.5 we have for every t ≥ Since X , X , Y and Y have mean 0 and variance 1 we may bound Since L is smaller than 1, lim t→∞ ||C t − C * ||∞ = . (4.30) Which concludes the proof of Proposition 3.3.

. Generalized solutions of PDEs
In this section we begin the proof of Theorem 3.4. We show that the copula process C t is a "weak generalized" solution of PDE (3.12) (we follow the naming used in [13] and [11], see also [15]). We assume A1 -A4. By H we denote the Hilbert space of weakly di erentiable functions, which together with their derivatives are square integrable.
where for xed t ≥ and xed copula C, B t (·, C) is a continuous linear functional on H (R ), given by the formula

Proof.
We base on results from [15]. We shift time t by 1, put σ i = and µ i = and apply Theorem 4.1 from [15].
Since for selfsimilar processes the copula process C t is constant, say C t = C for t ≥ , we get B (h, C) = . (4.33) To continue the proof of theorem 3.4 we have to improve the regularity of C. We apply Theorem 8.8 of [13]. In the notation used in [13], the divergence form of B looks like We restrict the domain to the smaller square Ωr = (Φ(−r), Φ(r)) , r > . We observe that, for any r > , on Ωr the coe cients a , , a , , a , , c and c are bounded. Furthermore a , , a , , a , are Lipschitz and B is strongly elliptic (4.40) Therefore Theorem 8.8 of [13] implies that C(u, v) belongs to H (Ωr) for any r > , thus it is weakly twice di erentiable on ( , ) and ful lls almost everywhere equation (3.11).
When C is weakly twice di erentiable so is the cumulative distribution function F(x, y) = C(Φ(x), Φ(y)). Furthermore If C ful lls almost everywhere equation (4.41), then F ful lls almost everywhere on R the equation (3.12) In such a way we conclude the proof of Theorem 3.4.
Remark 4.1. 1. Equation (3.11) can be derived from the Fokker-Planck equation (see [7]) But this requires the existence and twice-di erentiability of the density of the process (X t , Y t ). (3.11) and (3.12) can be derived without the assumption A4, see [22].

. Dominance and uniqueness
Theorem 4.9. Let A i , i = , , be bounded Lipschitz functions, |A i | < . If copulas C and C ful ll PDE (3.11) with A equal respectively to A and A and then copula C dominates in concordance ordering The proof of Theorem 3.5 follows from the maximum principle for elliptic partial di erential equations, see [13] or [11]. In more details: If copulas C and C ful ll PDE (3.11) with A equals respectively to A and A and A dominates ) is a sub-solution of the equation (3.11) with A equals to A with zero boundary condition. Indeed, since C is a copula its mixed derivative is almost everywhere nonnegative and for almost all (u, v) ∈ (o, ) we get Let δ be a minimum of U and let it be attained at the point (u , v ) We show that δ must be 0. If it happened that δ < then knowing that U is a Lipschitz function vanishing on the border of the unit square we would be able to select an enough big r that: 1. the point (u , v ) would belong to Ωr = (Φ(−r), Φ(r)) ; 2. the in mum of U on the complement of Ωr would be greater than δ/ . Next basing on the same arguments as in the previous section, we would apply Theorem 8.1 from [13] and would get Dividing both sides by negative δ we would get < / , a contradiction.
Hence U = C − C must be nonnegative, i.e. copula C dominates in concordance ordering. This concludes the proof of Theorem 3.5. To prove Corollary 3.6 we consider two copulas C(u, v) and D(u, v) which ful ll PDE (3.11) with the same A. Due to proved above Theorem 3.5 C dominates D and D dominates C and they have to be equal each other.
Let a copula C(x, y) be twice di erentiable in a weak sense and ful ll the equation (3.11) with A(x, y) bounded, |A| < , A and √ − A Lipschitz and A di erentiable with respect to the second variable. Due to Theorem 3.2 there exits a / -self-similar solution (X t , Y t ) t≥ of eq. (3.1) with coe cients A and B = √ − A . Let D be the copula of (X t , Y t ) for all t ≥ . Due to Theorem 3.4 copula D is a solution of the equation (3.11) with coe cient A(x, y). Hence Corollary 3.6 implies that C = D. Which means that C is a copula of the mentioned above a / -self-similar process (X t , Y t ) t≥ . Theorem 3.9 follows from Proposition 4.6, Corollary 4.7, "PDE characterization" Theorem 3.4 and the "uniqueness" Corollary 3.6. Proposition 4.6 and Corollary 4.7 imply that every cluster point (cluster copula) of the sequence C n = n e n s C C s , n = , , . . . is a copula of a 1/2-selfsimilar process. If we assume condition A4 then the cluster copulas ful ll the same PDE. Hence due to Corollary 3.6 they coincide. Which implies that the limit of the sequence Cn exists.

Examples . Gaussian copula
Let us recall that the Gaussian copula is de ned as follows: where Φρ is the joint distribution function of a bi-dimensional standard normal vector, with linear correlation coe cient ρ. For more details see [25] or [20] §4.3.1.
where t ≥ and ϕ (x, y, r) is a density of a bivariate normal distribution with correlation r and standard margins. Note that ϕ is bounded. For all (x, y) ∈ R and r ∈ (− , ) ϕ (x, y, r) ≤ π √ − r .

. FGM copula
The Farlie-Gumbel-Morgenstern copula is de ned as The partial derivatives of C are given by After substituting into eq. (3.11) we obtain We show that |A| is bounded by |a|/ for any a ∈ [− , ], so the maximum of |A| is attained to the point ( , ) (compare [16] Section 3.4). Indeed, we have an estimate for any x, y ∈ R and a ∈ [− , ] Since the right side of (5.2) is invariant with respect to change of sign of x, change of sign of y and symmetry (x, y) → (y, x) it is enough to show that for y ≥ x ≥ , a function is bounded by 1. We observe that the directional derivative is nonnegative for y ≥ x ≥ . Hence χ is nondecreasing along the hyperbola Since moreover χ(x, y) is positive for y ≥ x ≥ , we get Furthermore, since both nominator and denominator have bounded derivatives and for |a| < denominator is bounded from zero, A is di erentiable with bounded derivatives. Hence A and B(x, y) = − A(x, y) are Lipschitz. Therefore by Theorem 3.4 point 3 there exists a / -self-similar solution of eq. (3.1), with coecients A as above and B(x, y) = − A(x, y) , such that the copula of its initial values is FGM with parameter a ∈ (− , ).
The function g is called a generator of the copula Cg. For more details concerning Archimedean copulas, both strict and nonstrict, the reader is referred to [9,19,20,26].
Due to convexity, g is di erentiable at all but at most countably many points and the derivative may have jumps. Therefore, for simplicity, we restrict ourselves to twice di erentiable generators g. Then we have for t ∈ ( , ) g (t) < , g (t) ≥ . (5.4) We start with calculating the partial derivatives of C. In order to simplify the notation let us introduce auxiliary variable z = g − (g(u) + g(v)).
As we show in next subsections for some generators A ful ll assumptions A2 and A3, for some not.

. . Clayton copula
As a speci c example let us consider the following generator g(t) = α t −α − , α > . (5.10) This way we in fact obtain a member of the Clayton family of copulas (compare [9,20,26]), Since g (t) = −t −α− and g (t) = (α + )t −α− , we get We show that A is not bounded by 1. We apply the limit lim z→−∞ |z|Φ(z) ϕ(z) = , (5.13) which follows from the de l'Hospital rule: 14) We substitute x = z and y = z and take the limit as z tends to −∞.

Con ict of interest statement:
Authors state no con ict of interest.