Luis Fuentes García

Grüss-type bounds for covariances and the notion of quadrant dependence in expectation

We show that Grüss-type probabilistic inequalities for covariances can be considerably sharpened when the underlying random variables are quadrant dependent in expectation (QDE). The herein established covariance bounds not only sharpen the classical Grüss inequality but also improve upon recently derived Grüss-type bounds under the assumption of quadrant dependency (QD), which is stronger than QDE. We illustrate our general results with examples based on specially devised bivariate distributions that are QDE but not QD. Such results play important roles in decision making under uncertainty, and particularly in areas such as economics, finance, and insurance.

Gruss-Type Bounds for the Covariance of Transformed Random Variables

A number of problems in Economics, Finance, Information Theory, Insurance, and generally in decision making under uncertainty rely on estimates of the covariance between (transformed) random variables, which can, for example, be losses, risks, incomes, financial returns, and so forth. Several avenues relying on inequalities for analyzing the covariance are available in the literature, bearing the names of Chebyshev, Grüss, Hoeffding, Kantorovich, and others. In the present paper we sharpen the upper bound of a Grüss-type covariance inequality by incorporating a notion of quadrant dependence between random variables and also utilizing the idea of constraining the means of the random variables.

An Optimal Strategy for Maximizing the Expected Real-Estate Selling Price: Accept or Reject an Offer?

Motivated by a real-life situation, we put forward a model and then derive an optimal strategy that maximizes the expected real-estate selling price when one of the only two remaining buyers has already made an offer but the other one is yet to make. Since the seller is not sure whether the other buyer would make a lower or higher offer, and given no recall, the seller needs a strategy to decide whether to accept or reject the first-come offer. The herein derived optimal sellers strategy, which maximizes the expected selling price, is illustrated under several scenarios, such as independent and dependent offers by the two buyers, and for several parametric price distributions.

Convex combinations of quadrant dependent copulas

Abstract It is well known that quadrant dependent (QD) random variables are also quadrant dependent in expectation (QDE). Recent literature has offered examples rigorously establishing the fact that there are QDE random variables which are not QD. The examples are based on convex combinations of specially chosen QD copulas: one negatively QD and another positively QD. In this paper we establish general results that determine when convex combinations of arbitrary QD copulas give rise to negatively or positively QD/QDE copulas. In addition to being an interesting mathematical exercise, the established results are helpful when modeling insurance and financial portfolios.

Projective Normality of Special Scrolls

We study the projective normality of a linearly normal special scroll R of degree d and speciality i over a smooth curve X of genus g. We relate it with the Clifford index of the base curve X. If d ≥ 4g − 2i − Cliff(X) + 1, i ≥ 3 and R is smooth, we prove that the projective normality of the scroll is equivalent to the projective normality of its directrix curve of minimum degree.

The variance upper bound for a mixed random variable

ABSTRACT In this note, we derive upper bounds on the variance of a mixed random variable. Our results are an extension of previous results for unimodal and symmetric random variables. The novelty of our findings is that this mixed random variable does not necessarily need to be symmetric and is multimodal. We also characterize the cases when these bounds are optimal.

Simple models in finance: A mathematical analysis of the probabilistic recognition heuristic

It is well known that laypersons and practitioners often resist using complex mathematical models such as those proposed by economics or finance, and instead use fast and frugal strategies to make decisions. We study one such strategy: the recognition heuristic. This states that people infer that an object they recognize has a higher value of a criterion of interest than an object they do not recognize. We extend previous studies by including a general model of the recognition heuristic that considers probabilistic recognition, and carry out a mathematical analysis. We derive general closed-form expressions for all the parameters of this general model and show the similarities and differences between our proposal and the original deterministic model. Corresponding author: M. Egozcue Print ISSN 1753-9579 jOnline ISSN 1753-9587 Copyright

Bank's Equity-Asset Ratio Bounds Under Exchange Rate Risk: A Tool for Stress Testing

A large fraction of banks assets and debts in developing countries are usually denominated in foreign currencies (e.g. US dollars, euros). Therefore, fluctuations of the exchange rate is likely to affect their financial wealth. The equity-asset ratio is a usual measure of banks financial strength. As we will see, this ratio is very sensitive to exchange rate movements. In this paper, we establish some bounds of the expectation of the equity-asset ratio considering limited information of the underlying stochastic exchange rate. Specially, we are interested in finding the lower bound of the expectation of this ratio, that is knowing the worst case scenario of bank financial wealth. We also characterize the conditions under which the equity-ratio remains unchanged considering both direct and indirect effects of exchange rate movements. Our results could shed some light on the complex world of financial regulation and serve regulators as an additional tool for stress testing.

Winning Strategy in the Two Envelope Game with Partial Information

In this note, we study a winning strategy in the two envelope game. We assume the player only has partial information of the game. In particular, if the players knows the mean and variance of certain distribution, we show a winning strategy that assures a minimum average gain.

Revisiting Grüss’s Inequality: Covariance Bounds, QDE but not QD Copulas, and Central Moments

Since the pioneering work of Gerhard Gruss dating back to 1935, Gruss’s 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 Gruss’s 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.