We derive upper and lower bounds on the expectation of f(S) under dependence uncertainty, that is, when the marginal distributions of the random vector S = (S1,..., Sd) are known but their dependence ...structure is partially unknown. We solve the problem by providing improved Fréchet–Hoeffding bounds on the copula of S that account for additional information. In particular, we derive bounds when the values of the copula are given on a compact subset of 0, 1d, the value of a functional of the copula is prescribed or different types of information are available on the lower dimensional marginals of the copula. We then show that, in contrast to the two-dimensional case, the bounds are quasi-copulas but fail to be copulas if d > 2. Thus, in order to translate the improved Fréchet–Hoeffding bounds into bounds on the expectation of f(S), we develop an alternative representation of multivariate integrals with respect to copulas that admits also quasi-copulas as integrators. By means of this representation, we provide an integral characterization of orthant orders on the set of quasi-copulas which relates the improved Fréchet–Hoeffding bounds to bounds on the expectation of f(S). Finally, we apply these results to compute model-free bounds on the prices of multi-asset options that take partial information on the dependence structure into account, such as correlations or market prices of other traded derivatives. The numerical results show that the additional information leads to a significant improvement of the option price bounds compared to the situation where only the marginal distributions are known.
In this paper we address the problem of
α
-migrativity (for a fixed
α
) for semicopulas, copulas and quasi-copulas. We introduce the concept of an
α
-sum of semicopulas. This new concept allows us to ...completely characterize
α
-migrative semicopulas and copulas. Moreover,
α
-sums also provide a means to obtain a partial characterization of
α
-migrative quasi-copulas.
Baire category results for quasi–copulas Durante, Fabrizio; Fernández-Sánchez, Juan; Trutschnig, Wolfgang
Dependence modeling,
10/2016, Letnik:
4, Številka:
1
Journal Article
Recenzirano
Odprti dostop
The aim of this manuscript is to determine the relative size of several functions (copulas, quasi–
copulas) that are commonly used in stochastic modeling. It is shown that the class of all ...quasi–copulas that
are (locally) associated to a doubly stochastic signed measure is a set of first category in the class of all quasi–
copulas. Moreover, it is proved that copulas are nowhere dense in the class of quasi-copulas. The results are
obtained via a checkerboard approximation of quasi–copulas.
Geometry of discrete copulas Perrone, Elisa; Solus, Liam; Uhler, Caroline
Journal of multivariate analysis,
07/2019, Letnik:
172
Journal Article
Recenzirano
Odprti dostop
The space of discrete copulas admits a representation as a convex polytope, and this has been exploited in entropy-copula methods used in hydrology and climatology. In this paper, we focus on the ...class of component-wise convex copulas, i.e., ultramodular copulas, which describe the joint behavior of stochastically decreasing random vectors. We show that the family of ultramodular discrete copulas and its generalization to component-wise convex discrete quasi-copulas also admit representations as polytopes. In doing so, we draw connections to the Birkhoff polytope, the alternating sign matrix polytope, and their generalizations, thereby unifying and extending results on these polytopes from both the statistics and the discrete geometry literature.
This paper is mainly devoted to solving the functional equations of distributivity and conditional distributivity of increasing binary operations with the unit. Our investigations are motivated by ...distributive logical connectives and their generalizations used in fuzzy set theory. In particular, some assumptions (namely associativity and commutativity) and results about conditional distributivity of uninorms and triangular norms and triangular conorms were simplified. Moreover, some other properties e.g. componentwise convexity (concavity) for special operations from the class of quasi-copulas is considered.
We consider flipping transformations of multivariate aggregation functions and we investigate the closure of these transformations with respect to the class of aggregation functions with annihilator ...element equal to 0. Moreover, the consecutive application of flipping transformations is also discussed. The results present interesting connections both with quasi-copulas and with suitable modifications of the n-increasing property of a multivariate probability distribution function.
The fundamental best-possible bounds inequality for bivariate distribution functions with given margins is the Fréchet–Hoeffding inequality: If
H denotes the joint distribution function of random ...variables
X and
Y whose margins are
F and
G, respectively, then max(0,
F(
x)+
G(
y)−1)⩽
H(
x,
y)⩽min(
F(
x),
G(
y)) for all
x,
y in −∞,∞. In this paper we employ copulas and quasi-copulas to find similar best-possible bounds on arbitrary sets of bivariate distribution functions with given margins. As an application, we discuss bounds for a bivariate distribution function
H with given margins
F and
G when the values of
H are known at quartiles of
X and
Y.
We study a method, which we call a copula (or quasi-copula)
diagonal splice, for creating new functions by joining portions of two copulas (or quasi-copulas) with a common diagonal section. The ...diagonal splice of two quasi-copulas is always a quasi-copula, and we find a necessary and sufficient condition for the diagonal splice of two copulas to be a copula. Applications of this method include the construction of absolutely continuous asymmetric copulas with a prescribed diagonal section, and determining the best-possible upper bound on the set of copulas with a particular type of diagonal section. Several examples illustrate our results.
The aim of this paper is to choose diverse definitions of generalized logical connectives and to present them in a coherent order, from the weakest to the richest. Such a rich list of notions allows ...us to consider the problem of admissible aggregations in the presented classes of unary and binary operations. This gives a contribution to the discussion of tolerance analysis in soft computing, decision making, approximate reasoning, and fuzzy control. First, we present a short survey of the development of MV-logic's connectives. Next, we discuss postulates used for generalized logical connectives. Finally, we describe families of weak connectives and indicate the preservation of their properties by some aggregation functions.
The theory of copulas is by now a very well established one. Recently, larger classes of functions
C
:
0
,
1
n
→
0
,
1
, that are increasing in each variable and satisfy some conditions at the ...boundary (like quasi-copulas), have been the object of fruitful research. Several authors have considered the action of a class of transformations on some aggregation operators, as t-norms, copulas, quasi-copulas and so on. These simple transformations do not preserve in general all properties of copulas (or quasi-copulas): in particular, the fact that only some properties are preserved by these transformations, suggested the introduction of semi-copulas. The purpose of the present contribution is to give a fairly complete picture concerning such action on copulas and quasi-copulas; in particular, we prove results concerning inclusions and strict inclusions among these classes of operators, and those of their transforms.
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