Copulas with fractal supports

Abstract

Using an iterated function system, the authors construct families of copulas whose supports are fractals. In particular, they show that the members of one family have supports with arbitrary Hausdorff dimensions in the interval (1,2). They also employ those copulas to construct more general bivariate distribution functions with fractal supports.

Limiting dependence structures for tail events, with applications to credit derivatives

“…Observe that the hypothesis that r and s are continuous functions is a necessary one, since otherwise counterexamples such as copulas with fractal support, as considered by Fredricks et al (2005), can be constructed. Let T = (t ij ) be a square matrix, with nonnegative entries whose sum equals 1, determining the following subdivision of the unit square …”

“…Fredricks et al (2005) showed that for any copula C and any T = 1 there is a unique copula, C T , that depends only on T and satisfies T (C T ) = C T . Moreover, they showed that…”

Some Consequences of the Markov Kernel Perspective of Copulas

“…Since the star product of copulas is a natural generalization of the multiplication of doubly stochastic matrices and doubly stochastic idempotent matrices are fully characterizable (see [10,25]) the following result underlines how much more complex the family of idempotent copulas is (also see [12] for the original result without idempotence).…”

“…For general background on Iterated Function Systems with Probabilities (IFSP, for short), we refer to [16]. The IFSP construction of twodimensional copulas with fractal support goes back to [12] (also see [2]), for the generalization to the multivariate setting we refer to [30].…”

“…According to [12] there is exactly one copula A τ ∈ C , to which we will refer to as invariant copula, such that Moreover (again see [12]) the support …”