Copulas: a Review and Recent Developments (2007)

ution functions, see Section 2.1.1 and Nelsen et al. (2003), in order to model the dependencestructure between d random vectors (X 1 ;::: ;X m1 ),...,(X md¡1 +1;::: ;X md ),with m d = n using a d-dimensional copula (2 · d · n) to represent the dependencestructure between nonoverlapping marginals which are in general multivariate.The need to study relationships among nonoverlapping random vectors arises naturallyin a variety of circumstances. Such problems arise if one needs to build astochastic model in a situation where the information about the kind of dependenceand knowledge of certain marginal distributions is available. Typically, one has anidea about the dependence mechanism of nonoverlapping segments composing theportfolio, and the basic interest is to search a dependence structure between that segments.Examples are complex engineering systems, Jackson queueing network, controlof risk clustering, pricing and hedging sensitive instruments, pricing and hedgingbasket derivatives and structured products, credit portfolio management, credit andmarket risk measurement, etc. The basic result is the following.Sklar's Theorem (for copulas with multivariate marginals, Anjos and Kolev(2005a)). Let the continuous random variables X 1 ;::: ;X n are distributed into dnonoverlapping random vectors X l =(X ml¡1 +1;::: ;X ml ),withm l = P lj=1 n j, m 0 =0, l =1;::: ;d, 2 · d · n and n = n 1 + :::+ n d .LetH l be the marginal distributionof X l and C Hl be the associated copula of dimension n l , respectively. Then there existsan unique d-dimensional copula C d given byC d (w 1 ;::: ;w d )=P (W 1 · w 1 ;::: ;W d · w d ) (6)for (w 1 ;::: ;w d ) 2 [0; 1] d , representing the dependence structure between X 1 ;::: ;X d ,where W l = K l (H l (X l )) » U(0; 1) and K l is the Kendall distribution function of X l ,l =1;::: ;d. ¥One needs to impose restrictions on marginals H l in order C d to be a n-dimensionaldistribution function of (X 1 ;::: ;X n ). As Marco and Ruiz-Rivas (1992) showed, inthe simplest case, for an arbitrary d-dimensional copula those restrictions lead to maxin¯nitelydivisible marginals. The Nutshell copula's paradox discussed by Genest etal. (1995) is a consequence of such limitations.Problems of this kind arise if one needs to build a stochastic model in a situationwhere the information about the kind of dependence and knowledge of certainmarginal distributions is available. The method based on the theorem above, is motivatedby the following fact. Typically, one has an idea about the dependence mechanismof nonoverlapping segments composing the portfolio, and the basic interest isto search a dependence structure between that segments.The statement of the theorem can be considered as a version of the Sklar's theoremfor copulas with nonoverlapping multivariate marginals. If substitute d = n in (6),one gets (2), i.e. the conclusion of the Sklar's theorem (when the univariate marginalsare ¯xed or known).25