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Register a new userCharacterization of Extreme Copulas
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In this paper our aim is to characterize the set of extreme points of the set of all n-dimensional copulas (n > 1). We have shown that a copula must induce a singular measure with respect to Lebesgue measure in order to be an extreme point in the set of n-dimensional copulas. We also have discovered some sufficient conditions for a copula to be an extreme copula. We have presented a construction of a small subset of n-dimensional extreme copulas such that any n-dimensional copula is a limit point of that subset with respect to weak convergence. The applications of such a theory are widespread, finding use in many facets of current mathematical research, such as distribution theory, survival analysis, reliability theory and optimization purposes. To illustrate the point further, examples of how such extremal representations can help in optimization have also been included.
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For extreme value copulas with a known upper tail dependence coefficient we find pointwise upper and lower bounds, which are used to establish upper and lower bounds of the Spearman and Kendall correlation coefficients. We shown that in all cases the lower bounds are attained on Marshall--Olkin copulas, and the upper ones, on copulas with piecewise linear dependence functions.
We propose a new method for modeling the distribution function of high dimensional extreme value distributions. The Pickands dependence function models the relationship between the covariates in the tails, and we learn this function using a neural network that is designed to satisfy its required properties. Moreover, we present new methods for recovering the spectral representation of extreme distributions and propose a generative model for sampling from extreme copulas. Numerical examples are provided demonstrating the efficacy and promise of our proposed methods.
The objective in stochastic filtering is to reconstruct information about an unobserved (random) process, called the signal process, given the current available observations of a certain noisy transformation of that process. Usually X and Y are modeled by stochastic differential equations driven by a Brownian motion or a jump (or Levy) process. We are interested in the situation where both the state process X and the observation process Y are perturbed by coupled Levy processes. More precisely, L=(L_1,L_2) is a 2--dimensional Levy process in which the structure of dependence is described by a Levy copula. We derive the associated Zakai equation for the density process and establish sufficient conditions depending on the copula and $L$ for the solvability of the corresponding solution to the Zakai equation. In particular, we give conditions of existence and uniqueness of the density process, if one is interested to estimate quantities like P( X(t)>a), where a is a threshold.
Since the pioneering work of Gerhard Gruss dating back to 1935, Grusss inequality and, more generally, Gruss-type bounds for covariances have fascinated researchers and found numerous applications in areas such as economics, insurance, reliability, and, more generally, decision making under uncertainly. Gruss-type bounds for covariances have been established mainly under most general dependence structures, meaning no restrictions on the dependence structure between the two underlying random variables. Recent work in the area has revealed a potential for improving Gruss-type bounds, including the original Grusss bound, assuming dependence structures such as quadrant dependence (QD). In this paper we demonstrate that the relatively little explored notion of `quadrant dependence in expectation (QDE) is ideally suited in the context of bounding covariances, especially those that appear in the aforementioned areas of application. We explore this research avenue in detail, establish general Gruss-type bounds, and illustrate them with newly constructed examples of bivariate distributions, which are not QD but, nevertheless, are QDE. The examples rely on specially devised copulas. We supplement the examples with results concerning general copulas and their convex combinations. In the process of deriving Gruss-type bounds, we also establish new bounds for central moments, whose optimality is demonstrated.
We derive upper and lower bounds on the expectation of $f(mathbf{S})$ under dependence uncertainty, i.e. when the marginal distributions of the random vector $mathbf{S}=(S_1,dots,S_d)$ are known but their dependence structure is partially unknown. We solve the problem by providing improved FH bounds on the copula of $mathbf{S}$ that account for additional information. In particular, we derive bounds when the values of the copula are given on a compact subset of $[0,1]^d$, 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 FH bounds into bounds on the expectation of $f(mathbf{S})$, 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 FH bounds to bounds on the expectation of $f(mathbf{S})$. 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.
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