On Copula-Itô processes

Abstract We study the dynamics of the family of copulas {Ct}t≥0 of a pair of stochastic processes given by stochastic differential equations (SDE). We associate to it a parabolic partial differential equation (PDE). Having embedded the set of bivariate copulas in a dual of a Sobolev Hilbert space H1 (ℝ2)* we calculate the derivative with respect to t and the *weak topology i.e. the tangent vector field to the image of the curve t → Ct. Furthermore we show that the family {Ct}t≥0 is an orbit of a strongly continuous semigroup of transformations and provide the infinitesimal generator of this semigroup.


Introduction
There is a vast literature dealing with the borderline of the copula theory and stochastic processes. There are two main objectives, temporal dependence and spatial dependence. The rst one concerns the studies of dependencies over time: given a one-dimensional stochastic process (X t ) t∈T and n times t , . . . , tn ∈ T determine the copula of the random vector (X t , . . . , X tn ). The research in this direction was originated by seminal paper of Darsow, Nguyen and Olsen [7] and continued by many authors, amongst others Sempi [26], Bibbona et al. [2] and Schmitz [23]. The second one concerns the vector valued stochastic processes X t = (X t , . . . , X n t ), t ∈ T. The goal is to describe the evolution of interdependencies between X t , . . . , X n t in terms of copulas and copula processes. It is represented by papers by Sempi [25], Choe et al. [6] and Jaworski and Krzywda [12,13].
This paper belongs to the second category. The work here is strongly in uenced by the paper of Sempi ( [25]) who was studying the possibility of coupling two Wiener processes by using a given copula, and the research on copulas of self-similar processes in [12,13]. It generalized the results presented in [6].
Due to the celebrated Sklar's Theorem, one can split a study of a pair (X, Y) of random variables into two parts. First one can deal with X and Y separately and then with the interdependence between them. In this paper we apply this methodology to the study of a pair of stochastic processes. We assume that dynamics of each process is described by a stochastic di erential equation. Our objective is to describe the evolution of the interdependence. This is done in terms of a family of copulas C t , linking the corresponding elements of the both processes. Having embedded the set of bivariate copulas C into the dual to the Sobolev space H (R ) we calculate the vector eld tangent to the image of the family (C t ) t≥ . Moreover, under some additional assumptions implying the existence and uniqueness of of the solutions of SDE, we show that there exists a continuous semigroup Λ of transformations (endomorphisms) of the Cartesian product of nonnegative real numbers and the set of bivariate copulas R + × C Λ : R + −→ End(R + × C ) *Corresponding Author: Piotr Jaworski: Institute of Mathematics, University of Warsaw, Poland, Email: P.Jaworski@mimuw.edu.pl such that the graph (t, C t ) is an orbit of Λ. Having embedded the space of copulas into the dual to the Sobolev space H (R ) we calculate the in nitesimal generator of Λ.
Structure of the paper: First we shortly recall basics on copulas and the Sobolev space H (R ) and its dual. Then we introduce the underlying stochastic processes X t and X t and the copula process (C t ) t≥ . Next we show how to transform the stochastic di erential equations of X and X to stochastic di erential equations describing the processes of representers of the corresponding copulas C t (section 3). In section 4 we state our main results concerning the evolution of the family (C t ) t≥ . We discuss the di erentiability of C t with respect to t and provide a parabolic di erential equation for which the family C t serves as a weak solution. In the following (section 5) we construct the semigroup of transformations Λ describing the dependence of the evolution of the families C t on initial values and discuss its properties. The proofs are relegated to section 6.

Notation
We recall the following 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: We recall that every bivariate copula is a restriction to the unit square [ , ] of the cumulative distribution function of a vector (U , U ) of standard uniform random variables, i.e., for all u , u ∈ [ , ], C(u , u ) = Pr(U ≤ u , U ≤ u ) and for i = , , Pr(U i ≤ u i ) = u i . Such random variables U , U will be called representers of the copula C. Indeed, such a pair serves as a representative of the equivalence class of the pairs of random variables which are uniformly distributed on the unit interval (0,1) and admit C as their copula.
Due to Sklar's Theorem, the joint distribution function F of any pair (X, Y) of random variables de ned on the probability space (Ω, M, 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 continuous random variables, 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. In this paper we show that sometimes it is useful to consider the space of copulas with a more re ned topology (compare [27]) and a more re ned embedding into a Banach space. For more details about copula theory and some of its applications, we refer to ([5, 8, 14-17, 20]).
Note that the copulas are weakly di erentiable. We recall that a function ψ(x , x ), de ned on an open set U, is called a weak partial derivative of a function Ψ(x , x ) with respect to the ith variable, written for all in nitely many times di erentiable test functions with compact support h, h ∈ C ∞ c (U). To distinguish the classical partial derivatives from their weak generalizations, we will denote the rst ones by ∂ ∂v i and the second ones by D i . Since copulas are Lipschitz functions, they are di erentiable except possibly some set of Lebesgue measure 0. Therefore any measurable extension of the partial derivative is a weak derivative.
The set H (R ) consisting of weakly di erentiable functions h de ned on the real plane, such that h and its weak derivatives D h and D h are square integrable, endowed with a scalar product is a Hilbert space, an example of the Sobolev spaces. For more details the reader is referred for example to [3,9,11].
In the following we associate with every copula a continuous linear functional on H (R ) i.e. an element of the dual space H (R ) * C → [ , ] h(x, y)C(x, y)dxdy.
For our purposes the best suited is the *weak topology on H (R ) * . We say that a family of copulas (C t ) t≥ is di erentiable at a point t with respect to *weak topology on H (R ) * if and only if for every test function h ∈ H (R ) the limit exists.
Next when we are dealing not just with one one-parameter family but with a family of one-parameter families C, we require that it has a form of a ow, i.e. there exists a mapping Λ : R + −→ End(R + , C ), such that: 1. Λ is the identity mapping and the family of endomorphisms (Λ t ) t≥ with composition is a semigroup Λ t (Λs(u, C)) = Λ t+s (u, C).
2. Graphs of the families from C are images of orbits of Λ t → Λ t (u, C).
the tangent vectors to graphs of the families from C are described in terms of the in nitesimal generator of Λ.

Preliminaries
We consider a probability space (Ω, M, P), with ltration F = {F t } t≥ and a pair of randomly correlated Wiener processes W = (W t ) t≥ and W = (W t ) t≥ . We assume: A1. W and W are adapted to the ltration F and the increments W i t+h − W i t , i = , , are all independent of every event in F t for t ≥ and h > .
Let F adapted path-continuous processes (X , X ) = (X t , X t ) t≥ , be solutions of the following system of stochastic di erential equations: We understand the above as integral equations: where A2. µ i (t, x) and σ i (t, x) are continuous functions de ned on [ , +∞) × (−∞, +∞), continuously di erentiable with respect to x with bounded derivatives For more details the reader is referred to the extensive literature on SDE, for example [1,4,10,18,21,24]. Furthermore we assume: A3. The quadratic covariation of X and X can be described by a deterministic function of X and X d X , where |A| is a bounded by 1 and the product A4. The univariate distribution functions F i (t, x) = P(X i t ≤ x) are di erentiable with respect to t, for t > , right-side di erentiable at t = and twice di erentiable with respect to x, with positive rst derivatives with respect to x. The derivatives are continuous with respect to both variables and "regular" at ±∞, i.e. there exists ε > such that for i = , , almost uniformly with respect to t: lim Let C t be a corresponding copula process We set for i = , Obviously U ,t and U ,t are uniformly distributed and are the representers of the copula C t . The processes of representers U and U are also called the copula uniform marginal processes.
Since the assumption A4. allows us to apply the Ito's Lemma, there exist functions ν i (t, u) such that Indeed from Ito's Lemma we get for i = , Since by A4 the univariate distribution functions F i are strictly increasing, the above de nes ν's inside the unit interval. We extend ν's by 0. For u ≤ and u ≥ we set The last part of A4 and A2 imply that constructed in such a way ν i,j are continuous functions on [ , +∞) × R. Hence they are bounded on compact subsets of t i.e. on [ , T] × R. Moreover, since F i have di erentiable inverses, the volatilities ν i, are di erentiable with respect to x on the whole real line, except perhaps at the two points 0 and 1. Note where Drift and volatility of SDE for representers are closely related.
and for u ∈ ( , ) The proof of the proposition is provided in section 6.1.
In the example below we show that the volatilities ν i, (t, x) need not to be di erentiable at x = , .
Example 3.1. The Gaussian copula. Let (X , X ) be a shifted by 1 two-dimensional Wiener process. Since for every t ≥ the pair (X t , X t ) has a bivariate normal distribution and the correlation is constant, the corresponding copula is a Gaussian one (see [8] section 6.7 for more details). Such a process ful lls assumptions A1 -A4 with the following parameters and initial values: For u ∈ ( , ) we have where Φ and φ are the distribution function and the density of the standard normal probability law N( , ).
Furthermore both functions Φ − (u) and φ Φ − (u) are regularly varying at + and − . The rst one is slowly varying, while the second has index of variation equal 1. For more details the reader is referred to section 6.2. Note that for i, j = , , and, although ν i,j (t, u) are not di erentiable at 0 and 1, they are regularly varying with index of variation 1.

The Copula processes
We have de ned a one-parameter family of copulas (C t ) t≥ , i.e. a curve:

Problem:
Choose an embedding of C into a topological linear space such that the embedded family is not only continuous but also di erentiable with respect to the parameter t.
We provide the solution of this problem by embedding the space of copulas into the space of continuous linear functionals on the Hilbert space of Sobolev functions on the real plane H (R ).
The embedding j : is constructed in the following way h(x, y)C(x, y)dxdy.
Note that since copulas are bounded by 1, we have for every h ∈ H (R ) h(x, y) dxdy h(x, y) dxdy Hence j(C) is indeed a continuous linear functional. Furthermore the mapping is di erentiable with respect to the *weak topology on H (R ) * (for t = we take the right-side derivative).
where for xed t ≥ and xed copula C, B t (·, C) is a continuous linear functional on H (R ), given by a formula The proof is provided in section 6.3.
Since B t is uniformly bounded on bounded sets of t, j(C t ) is locally Lipschitz in respect to the strong topology on H (R ) * .

Theorem 4.2. For any T > and s, t ∈ [ , T] there exists a constant L T such that for all h ∈ H
The proof is provided in section 6.3.
When the copula C is twice di erentiable, the above can be restated in terms of partial di erential equations. Applying integration by parts with respect to respectively the rst and the second variable, we get: Therefore, using the terminology of the theory of partial di erential equations, we may say that the family C t , de ned by (10), is a weak solution ( [9]) of a parabolic di erential equation Note that equation (30) was already studied in literature, but under more demanding assumptions. In [6] it was proved for copulas C with twice di erentiable density, three times di erentiable marginals F i and the interdependence given by a quadratic correlation function A(t, x , x ), introduced in A3, being a product In [12] a stationary version of (30) was derived for twice di erentiable copulas and Brownian margins.
To illustrate the possible applications of Theorem 4.1 we provide the following example concerning comonotonic random processes. We recall that random variables are comonotonic when they admit a singular copula M as their copula For more details the reader is referred to [8] The proof is based on the study of the dynamic of ρ(C t ), where ρ is the Spearman rank correlation coe cient. It is provided in subsection 6.4.

A semigroup approach to copula processes
The theory of semigroups of operators plays an important role in two domains which were mentioned in the previous sections, di usion processes described by stochastic di erential equations and parabolic partial di erential equations (see for example [9,21]). We show that it might be applied to the study of copula processes as well. To do it, we have to add two more assumptions which will ensure the existence and uniqueness, necessary to construct a semigroup of transformations.

A5. The functions σ A and
A6. The functions σ and σ do not vanish and the supremum of |A| is less than 1.
such that C is the copula of (X ,s,C , X ,s,C ). The pair of stochastic processes (X ,s,C , X ,s,C ) is law-unique.
The proof is provided in section 6.5.
We denote the copula corresponding to the pair (X ,s,C t , X ,s,C t ) by C s,C t . Since the pair (X ,s,C , X ,s,C ) is law-unique, the copula process (C s,C t ) t≥ is well de ned. It does not depend on the choice of the probability space, Wiener processes and the pair (X ,s,C , X ,s,C ), as long as the assumptions of the theorem are ful lled.
Moreover, since the restricted process X ,s,C t +t , X ,s,C t +t t≥ , t > , coincides with the process with parameters shifted by t + s and initial copula equal copula of the pair X ,s,C t , X ,s,C t (i.e. C s,C t ) we get a chain rule for copulas The restricted copula process C s,C t +t t≥ , t > , coincides with the copula process with parameters shifted by t + s and initial copula equal copula C s,C t . Now we are in a position to de ne a semigroup of transformations Λ. To make our system "autonomous" we add one more dependent variable. Instead of a process (C t ) t≥ , we consider (t, C t ) t≥ . We set Here End denotes the set of endomorphisms. The above chain rule for copulas implies that Λ is a semigroup. Indeed: . Note that the graph of the copula process C t corresponding to the pair (X , X ) is an orbit of Λ. Indeed We show that that local properties of Λ can be described in terms of the bilinear form B introduced by (27). In order to be able to apply the concept of di erentiability, we imbed our state space into a Banach space where j is given by formula (24). We select the *weak topology on (H ) * . Following the notation used for semigroups of linear operators we introduce an in nitesimal generator L It shows that the choice of the embedding j and the *weak topology ensures that (id, j) • Λ t (s, C) is di erentiable for all (s, C), i.e. D(L) = R + × C .

Theorem 5.2.
With respect to the topology induced by imbedding j : C → (H (R )) * , Λ is a strongly continuous semigroup of transformations. Moreover the generator L is a mapping given by where for a test function h ∈ H (R ) The proof is provided in section 6.5.
Since the graph of the copula process C t corresponding to the pair (X , X ) is an orbit of Λ Λ t ( , C ) = (t, C t ), the tangent vector to the curve t → j(C t ) and the generator of the semigroup coincide.

Proofs and auxiliary results . Proof of Proposition 3.1
Since both drifts ν j, are bounded, Since the expected values of all U j,t are the same, we get for s > ν j, (s, u)du = .
Next, we calculate the characteristic function of the random variable U j,t , t ≥ .
Since ν j, are bounded, We divide (40) by z and and apply integration by parts to the rst integral. We get for all z ∈ R: Hence for all u ∈ [ , ] and s ≥ u ν j, (s, ξ )dξ = ν j, (s, u) .
Since ν j, are continuous and ν j, are di erentiable for u ∈ ( , ), the above implies for s ≥ and u ∈ ( , ).

. A note on regular variation
We recall that a function U : ( , +∞) → ( , +∞) is regularly varying at +∞ with index ρ ∈ [−∞, +∞] if for every x > the limit lim t→+∞ U(tx) U(t) exists and equals x ρ , where (compare [22] De nition 2.1 and section 2.4). The variation with index 0 is also called the slow variation, while variation with index +∞ -the rapid variation. The above can be extended for functions F : ( , ) → (−∞, +∞). We say that F is regularly varying at + or respectively − with index ρ ∈ [−∞, +∞] if for every Let Φ, Φ − and φ denote the distribution function, the quantile function and the density of the standard normal distribution N( , ).
Proof. First we show that the auxiliary function U, is rapidly varying at +∞. Indeed applying the de l'Hospital rule we get Hence the inverse function U − is slowly varying at +∞ (see [22] Proposition 2.6.v). Since for y ∈ ( , ) Φ − is slowly varying at both ends of the unit interval.

Lemma 6.2. The composition of the density φ and the quantile function Φ − : ( , ) → (−∞, +∞) is regularly varying at + and − with index 1. For x
Proof. We apply the de l'Hospital rule and the previous lemma. Since This nishes the proof because φ is an even function and

. Proof of Theorems 4.1 and 4.2
We start with showing that assumption A3 implies that, although the derivatives of ν , might be unbounded, for xed v and s > the function has nite L norm. Furthermore the assumption A4 implies that there exists a constant M such that for all x ∈ R and s ∈ [ , T]

Lemma 6.3. For any xed T > there exists a constant M T such that for all s ∈ [ , T] and v ∈ ( , )
Applying the above we get: Now we switch to the bivariate case.
where for t ≥ and a copula C Applying an operator like approach, we can abbreviate the above as Next we substitute as H a convolution of an in nitely many times di erentiable function with compact support h and an indicator function κ(u )κ(u ) Note that H is in nitely many times di erentiable.
Furthermore note that for every u = (u , u ) Replacing h by its derivative we get ∂H ∂u j (u) ≤ ∂h ∂v j L and Using the Fubini Theorem, Sklar Theorem and the boundary properties of copulas we get for t ≥ Let Bs denote the integrand in formulas (46) and (47) multiplied by − . Since P(U j,s ≤ ) = , in a similar way as above, we obtain for s ≥ Now we apply the chain rule for conditional expectations Next we integrate by parts.
Now we show that the domain of integration can be restricted to the unit square. Since for x ≥ and s ≥ and ν i, are vanishing at the ends of the unit interval we have In a similar way we obtain To nish the proof we observe that Furthermore due to Lemma 6.3 we may apply once more the Fubini Theorem.
Applying the above to (56) we get We extend the bilinear form B t for Sobolev test functions. We set for h ∈ H We recall two useful facts about Sobolev functions from H . We start with absolutely continuity on lines (ACL) of Sobolev functions (see for example [19] §1.1.3.). If a function is in H (R ), then, possibly after modifying the function on a set of measure zero, the restriction to almost every line parallel to the coordinate directions in R is absolutely continuous; what is more, the classical derivative along the lines that are parallel to the coordinate directions are in L .
Therefore when h ∈ H (R ) has a compact support, then The second property is the possibility of approximation of Sobolev functions with compact supports by smooth functions with compact supports as well. We x a cut-o function Ψ.
Let h be a Sobolev function from H (R ). To "bound" the support, we set Basing on the above, we show that the extended bilinear form B t for xed copula C is a bounded linear mapping of test functions h. : Proof. We start with the following estimate: Since copulas are nondecreasing Lipschitz functions with Lipschitz constant 1, . Hence due to Hölder-Schwarz inequality ([9] Appendix B2) ≤ ||h|| H (R ) · ||ν , (t, ·)|| L ∞ + ||ν , (t, ·)|| L ∞ .
The estimation of the last summand is more complicated. We invert the transformations from the proof of Theorem 6.1 and then apply Hölder-Schwarz inequality. Let U , U be representers of the copula C and Ψ be the cut-o function de ned by formula (65).
[ , ] Putting the above three estimates together we obtain This implies the following uniform bound. For all t ∈ [ , T] we have where : Since ν i,j (t, v) are continuous functions the bound L T is well-de ned.
Since Bs(·, C) are uniformly bounded linear functionals on H , Theorem 6.1 remains valid for Sobolev test functions.
Proof. We x a test function h from H (R ). First to "bound" the support, we set All three terms converge as k → ∞. Indeed: for all t ≥ [ , ] h Similarly, basing on Lemma 6.4 with T = t we get

Lemma 6.5. Let the test function h belong to H (R ), then the function
Proof.
By assumption the processes X t and X t are path-continuous. Since their distribution functions F j (t, x) are continuous, the random variables U j,t = F j (t, X j t ) are path-continuous as well. Now, since the convolutions H are continuously di erentiable and the coecients L i,j are continuous, the composed stochastic process is path-continuous. Since it is bounded, we apply the Lebesgue dominated convergence theorem Which implies that B h (t) is continuous at each t ≥ .
Step 2. The general case h ∈ H (R ). Let h k , k = , , . . . , be a sequence of C ∞ functions with compact supports which approximate Almost uniform convergence implies the continuity.
To conclude the proofs of the theorems, one has to observe that Corollary 6.1 and Lemma 6.5 imply Theorem 4.1, and the same Corollary and Lemma 6.4 imply Theorem 4.2.

. Comonotonicity
The study of the form B t simpli es for singular copulas with support reduced to a union of segments. Especially for the copula having support equal to the diagonal section of the unit square. Note that the weak derivatives of M are discontinuous.
Lemma 6.6. For all t ∈ [ , +∞) and h ∈ H (R ) Proof. Applying (81), we obtain: Since the functions ν j, vanish at v j = and at v j = , we get: Note that, since |A C (t, v , v )| is bounded by 1 (see A3) the quadratic form is nonnegative. Furthermore it vanishes only when or or ν , (t, v) = ν , (t, v) = .
We recall that the Spearman ρ is given by a formula ρ(C) = [ , ] C(v , c )dv dv − .
the range of ρ is [− , ] and the maximum is attained only at C = M.
We apply Theorem 4.1 for a test function h equal 1 on the unit square We get

. Semigroups
In this section we provide proofs of Theorems 5.1 and 5.2, i.e. we show how to construct the bivariate stochastic processes (X ,s,C , X ,s,C ) and the generator of the semigroup Λ.
Proof of Theorem 5.1. Let C be a bivariate copula and s a nonnegative real number. It is well-known that for a bivariate distribution function F(x , x ) = C(F (s, x ), F (s, x )) there exists a pair of random variables (Y ,s,C , Y ,s,C ) de ned on a probability space Ω s,C , M s,C , P s,C such, that their distribution function equals F. Next letW = (W ,W ) be a standard two-dimensional Wiener process de ned on the probability space (Ω W , M W , P W ) and let F W denotes the natural ltration ofW.
We extend the random variables (Y ,s,C , Y ,s,C ) and the Wiener processW to the Cartesian product of probability spaces Ω W × Ω s,C , σ(M W × M s,C ), P W × P s,C .
Obviously the extensions, which will be denoted by the same symbols, are independent. We extent the ltration F as well F ext t = σ{V × W : V ∈ F W t , W ∈ M s,C }.
We de ne the bivariate stochastic process (X ,s,C t , X ,s,C t ) t≥ as a F ext adapted path-continuous solution of the following system of stochastic di erential equations: dX t = µ (t + s, X t )dt + σ (t + s, X t )dW t , dX t = µ (t + s, X t )dt + σ (t, +sX t )A(t + s, X t , X t )dW t (90) + σ (t + s, X t )B(t + s, X t , X t )dW t , with initial values Y ,s,C , Y ,s,C . The existence of the solution is ensured by the assumptions A2. and A5.. Indeed, µ , µ , σ , Aσ and Bσ are Lipschitz functions, hence of bounded growth and ful ll the assumptions of the theorems on the existence of the solutions of SDE (see for example [18]