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Ultramodularity and copulas. (English) Zbl 1371.62050

The authors study ultramodular binary copulas and characterize the additive generators of Archimedean ultramodular binary copulas. M. Marinacci and L. Montrucchio [Math. Oper. Res. 30, No. 2, 311–332 (2005; Zbl 1082.52006)] showed that ultramodularity of real functions is a stronger verson of both convexity and supermodularity. The ultramodularity of a copula stands for the mutually stochastically decreasing with respect to each other of two random variables.
The \(n\)-ary aggregation function \(A: [0,1]^n \to [0, 1]\) (monotone non-decreasing with two boundary conditions \(A(0, 0, \dots, 0)=0\), \(A(1, 1, \dots, 1) =1)\) is ultramodular if for all \(x, y, z\in [0, 1]^n\) with \(x+y+z\in [0,1]^n\), \[ A(x+y+z)+A(x) \geq A(x+y) + A(x+z). \] J. H. B. Kemperman [Nederl. Akad. Wet., Proc., Ser. A 80, 313–331 (1977; Zbl 0384.28012)] characterized that an \(n\)-ary function is supermodular if and only if each of its two-dimensional sections is supermodular; Marinacci and Montrucchio [Zbl 1082.52006] stated that an \(n\)-ary function is ultramodular if and only if it is supermodular and each of its one-dimensional sections is convex. These two results show that there are equivalent ways to characterize ultramodularity by
(i) ultramodularity on each two-dimensional section of the function or by
(ii) supermodularity on each two-dimensional section and convexity on each one-dimensional section.
Well-known examples of supermodular aggregation functions are copulas, and each ultramodular copula is negative quadrant dependent. The authors [Inf. Sci. 181, No. 19, 4101–4111 (2011; Zbl 1258.03082)] characterized ultramodularity by the supermodularity from the composite function on each monotone non-decreasing supermodular functions.
Section 1 and Section 2 give a brief introduction and backgrounds on supermodularity and ultramodularity. Section 3 focuses on ultramodular binary copulas. Note that each associate copula is an ordinal sum of Archimedean copulas. Therefore each associative ultramodular copula is a trivial sum of Archimedean copulas. Theorem 3.1 shows that a twice differentiable Archimedean copula is ultramodular if and only if \(1/t^\prime\) is a convex function for the additive generator \(t: [0, 1] \to [0, \infty]\), Theorem 3.5 proves that all the one-dimensional sections of an Archimedean copula are concave if and only if \(t^{\prime}(0)=\infty\), \(t^{\prime}\) is finite on \((0, 1]\) and \(1/t^{\prime}\) is concave. Theorem 3.7 addresses twice differentiable horizontal or vertical generators \(f\). Convexity on \(1/f^{\prime}\) leads to ultramodularity of the following copulas \[ C_f(x, y) =\begin{cases} 0 & \text{if \(x=0\),}\\ x \cdot f^{-1}(\min ( (f(y)/x), f(0)) & \text{otherwise}. \end{cases} \]
\[ C^f(x, y) =\begin{cases} 0 & \text{if \(y=0\),}\\ y \cdot f^{-1}(\min ( (f(x)/y), f(0)) & \text{otherwise}. \end{cases} \]
Section 4 is devoted to construct copulas from Theorem 2.8, and Theorem 4.1 gives a copula from a continuous ultramodular aggregation function and 2-ary copulas as well as continuous monotone non-decreasing functions. The copulas constructed by Theorem 4.1 are non-symmetric. Example 4.2 lists 4 examples from literatures. The authors expect to construct higher dimensional copulas from this approach.
It would be more interesting to construct ultramodular copulas with applications in financial economics and physics among other scientific fields which are not supermodular.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
26B25 Convexity of real functions of several variables, generalizations
26B35 Special properties of functions of several variables, Hölder conditions, etc.
62E10 Characterization and structure theory of statistical distributions
60E05 Probability distributions: general theory
62H10 Multivariate distribution of statistics
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