16,677 research outputs found

    Measuring the interactions among variables of functions over the unit hypercube

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    By considering a least squares approximation of a given square integrable function f ⁣:[0,1]nRf\colon[0,1]^n\to\R by a multilinear polynomial of a specified degree, we define an index which measures the overall interaction among variables of ff. This definition extends the concept of Banzhaf interaction index introduced in cooperative game theory. Our approach is partly inspired from multilinear regression analysis, where interactions among the independent variables are taken into consideration. We show that this interaction index has appealing properties which naturally generalize the properties of the Banzhaf interaction index. In particular, we interpret this index as an expected value of the difference quotients of ff or, under certain natural conditions on ff, as an expected value of the derivatives of ff. These interpretations show a strong analogy between the introduced interaction index and the overall importance index defined by Grabisch and Labreuche [7]. Finally, we discuss a few applications of the interaction index

    The idempotent Radon--Nikodym theorem has a converse statement

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    Idempotent integration is an analogue of the Lebesgue integration where σ\sigma-additive measures are replaced by σ\sigma-maxitive measures. It has proved useful in many areas of mathematics such as fuzzy set theory, optimization, idempotent analysis, large deviation theory, or extreme value theory. Existence of Radon--Nikodym derivatives, which turns out to be crucial in all of these applications, was proved by Sugeno and Murofushi. Here we show a converse statement to this idempotent version of the Radon--Nikodym theorem, i.e. we characterize the σ\sigma-maxitive measures that have the Radon--Nikodym property.Comment: 13 page

    Representation of maxitive measures: an overview

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    Idempotent integration is an analogue of Lebesgue integration where σ\sigma-maxitive measures replace σ\sigma-additive measures. In addition to reviewing and unifying several Radon--Nikodym like theorems proven in the literature for the idempotent integral, we also prove new results of the same kind.Comment: 40 page

    Stochastic domination for the Ising and fuzzy Potts models

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    We discuss various aspects concerning stochastic domination for the Ising model and the fuzzy Potts model. We begin by considering the Ising model on the homogeneous tree of degree dd, \Td. For given interaction parameters J1J_1, J2>0J_2>0 and external field h_1\in\RR, we compute the smallest external field h~\tilde{h} such that the plus measure with parameters J2J_2 and hh dominates the plus measure with parameters J1J_1 and h1h_1 for all hh~h\geq\tilde{h}. Moreover, we discuss continuity of h~\tilde{h} with respect to the three parameters J1J_1, J2J_2, hh and also how the plus measures are stochastically ordered in the interaction parameter for a fixed external field. Next, we consider the fuzzy Potts model and prove that on \Zd the fuzzy Potts measures dominate the same set of product measures while on \Td, for certain parameter values, the free and minus fuzzy Potts measures dominate different product measures. For the Ising model, Liggett and Steif proved that on \Zd the plus measures dominate the same set of product measures while on \T^2 that statement fails completely except when there is a unique phase.Comment: 22 pages, 5 figure

    Review of modern numerical methods for a simple vanilla option pricing problem

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    Option pricing is a very attractive issue of financial engineering and optimization. The problem of determining the fair price of an option arises from the assumptions made under a given financial market model. The increasing complexity of these market assumptions contributes to the popularity of the numerical treatment of option valuation. Therefore, the pricing and hedging of plain vanilla options under the Black–Scholes model usually serve as a bench-mark for the development of new numerical pricing approaches and methods designed for advanced option pricing models. The objective of the paper is to present and compare the methodological concepts for the valuation of simple vanilla options using the relatively modern numerical techniques in this issue which arise from the discontinuous Galerkin method, the wavelet approach and the fuzzy transform technique. A theoretical comparison is accompanied by an empirical study based on the numerical verification of simple vanilla option prices. The resulting numerical schemes represent a particularly effective option pricing tool that enables some features of options that are depend-ent on the discretization of the computational domain as well as the order of the polynomial approximation to be captured better

    A Possibilistic and Probabilistic Approach to Precautionary Saving

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    This paper proposes two mixed models to study a consumer's optimal saving in the presence of two types of risk.Comment: Panoeconomicus, 201

    Rating and ranking of multiple-aspect alternatives using fuzzy sets

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    A method is proposed to deal with multiple-alternative decision problems under uncertainty. It is assumed that all the alternatives in the choice set can be characterized by a number of aspects, and that information is available to assign weights to these aspects and to construct a rating scheme for the various aspects of each alternative. The method basically consists of computing weighted final ratings for each alternative and comparing the weighted final ratings. The uncertainty that is assumed to be inherent in the assessments of the ratings and weights is accounted for by considering each of these variables as fuzzy quantities, characterized by appropriate membership functions. Accordingly, the final evaluation of the alternatives consists of a degree of membership in the fuzzy set of alternatives ranking first. A practical method is given to compute membership functions of fuzzy sets induced by mappings, and applied to the problem at hand. A number of examples are worked out. The method is compared to another one proposed by Kahne who approaches the problem probabilistically
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