47 research outputs found

    Decomposition approaches to integration without a measure

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    Extending the idea of Even and Lehrer [3], we discuss a general approach to integration based on a given decomposition system equipped with a weighting function, and a decomposition of the integrated function. We distinguish two type of decompositions: sub-decomposition based integrals (in economics linked with optimization problems to maximize the possible profit) and super-decomposition based integrals (linked with costs minimization). We provide several examples (both theoretical and realistic) to stress that our approach generalizes that of Even and Lehrer [3] and also covers problems of linear programming and combinatorial optimization. Finally, we introduce some new types of integrals related to optimization tasks.Comment: 15 page

    On the Diversification Effect in Solvency II for Extremely Dependent Risks

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    In this article, we investigate the validity of diversification effect under extreme-value copulas, when the marginal risks of the portfolio are identically distributed, which can be any one having a finite endpoint or belonging to one of the three maximum domains of attraction. We show that Value-at-Risk (V@R) under extreme-value copulas is asymptotically subadditive for marginal risks with finite mean, while it is asymptotically superadditive for risks with infinite mean. Our major findings enrich and supplement the context of the second fundamental theorem of quantitative risk management in existing literature, which states that V@R of a portfolio is typically non-subadditive for non-elliptically distributed risk vectors. In particular, we now pin down when the V@R is super or subadditive depending on the heaviness of the marginal tail risk. According to our results, one can take advantages from the diversification effect for marginal risks with finite mean. This justifies the standard formula for calculating the capital requirement under Solvency II in which imperfect correlations are used for various risk exposures

    The Geometry and Calculus of Losses

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    Statistical decision problems lie at the heart of statistical machine learning. The simplest problems are binary and multiclass classification and class probability estimation. Central to their definition is the choice of loss function, which is the means by which the quality of a solution is evaluated. In this paper we systematically develop the theory of loss functions for such problems from a novel perspective whose basic ingredients are convex sets with a particular structure. The loss function is defined as the subgradient of the support function of the convex set. It is consequently automatically proper (calibrated for probability estimation). This perspective provides three novel opportunities. It enables the development of a fundamental relationship between losses and (anti)-norms that appears to have not been noticed before. Second, it enables the development of a calculus of losses induced by the calculus of convex sets which allows the interpolation between different losses, and thus is a potential useful design tool for tailoring losses to particular problems. In doing this we build upon, and considerably extend existing results on MM-sums of convex sets. Third, the perspective leads to a natural theory of ``polar'' loss functions, which are derived from the polar dual of the convex set defining the loss, and which form a natural universal substitution function for Vovk's aggregating algorithm.Comment: 65 pages, 17 figure

    Fuzzy Esscher changes of measure and copula invariance in Lévy markets

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    In the context of a multidimensional exponential Lévy market, we focus on the Esscher change of measure and suggest a more flexible tool allowing for a fuzzy version of the standard Esscher transform. Motivated both by the empirical incompatibility of market data and the analytical form of the standard Esscher transform (see [8]) and by the desire to introduce a pricing technique under incompleteness conditions, we detect the impact of fuzziness in terms of measure change function and in contingent claims' pricing. In a multidimensional setting the fuzzy Esscher transform is a copula whose invariance, under margins' transformations induced by a change of measure, is investigated and connected to the notion of the absence of arbitrage opportunities. We highlight how Esscher transform, primarily used in pricing techniques, preserves the invariance of the aggregation operator and it can be generalized to the fuzzy version assuming that the measurable functions defining the Choquet marginal integrals are increasing. Furthermore, the empirical evidence seems to suggest that a weaker concept of invariance may be more suitable, i.e. the ε-measure invariance, coherent with the Esscher fuzzy copula tool. An empirical experiment for our model will make clear how this blurring technique fits the market data

    Quantum Loewner Evolution

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    What is the scaling limit of diffusion limited aggregation (DLA) in the plane? This is an old and famously difficult question. One can generalize the question in two ways: first, one may consider the {\em dielectric breakdown model} η\eta-DBM, a generalization of DLA in which particle locations are sampled from the η\eta-th power of harmonic measure, instead of harmonic measure itself. Second, instead of restricting attention to deterministic lattices, one may consider η\eta-DBM on random graphs known or believed to converge in law to a Liouville quantum gravity (LQG) surface with parameter γ[0,2]\gamma \in [0,2]. In this generality, we propose a scaling limit candidate called quantum Loewner evolution, QLE(γ2,η)(\gamma^2, \eta). QLE is defined in terms of the radial Loewner equation like radial SLE, except that it is driven by a measure valued diffusion νt\nu_t derived from LQG rather than a multiple of a standard Brownian motion. We formalize the dynamics of νt\nu_t using an SPDE. For each γ(0,2]\gamma \in (0,2], there are two or three special values of η\eta for which we establish the existence of a solution to these dynamics and explicitly describe the stationary law of νt\nu_t. We also explain discrete versions of our construction that relate DLA to loop-erased random walk and the Eden model to percolation. A certain "reshuffling" trick (in which concentric annular regions are rotated randomly, like slot machine reels) facilitates explicit calculation. We propose QLE(2,1)(2,1) as a scaling limit for DLA on a random spanning-tree-decorated planar map, and QLE(8/3,0)(8/3,0) as a scaling limit for the Eden model on a random triangulation. We propose using QLE(8/3,0)(8/3,0) to endow pure LQG with a distance function, by interpreting the region explored by a branching variant of QLE(8/3,0)(8/3,0), up to a fixed time, as a metric ball in a random metric space.Comment: 132 pages, approximately 100 figures and computer simulation
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