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By Grossi M.

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For each discretetime horizon {t, . . , T }, with t = T − 1, T − 2, . . , cT (·, ·) measures the distance between a pair of scenarios on the whole time horizon {1, . . , T }. If the minimum is attained at kt∗ , then the scenario i ∗ is deleted and ∗ j ∗ ∈ arg min∗ cT ξ i ξ j , (13) j =i is the aggregating scenario, with i ∗ , j ∗ ∈ D(kt∗ ). The optimal redistribution rule for the probabilities is defined, according to the results in [GKHR03], as follows: k π˜ kτ,τ +1 j∗ , j∗ kτ +1, j ∗ = πkτ, j ∗ k ∗ i + πkτ,τ +1, , i∗ (14) for τ = t, .

For each time series the daily log-price variations have been calculated and considered as input for the scaling analysis. The lags a and b have been taken in the set of time horizons A = {1, 2, . . 01. The scaling surfaces are obtained by filtering only those estimated H0’s for which the corresponding diameter (7) is below the critical value of the Smirnov statistics. Scaling Laws in Stock Markets. An Analysis of Prices and Volumes 39 Fig. 1. Scaling surface of an sBm. Table 1. Analyzed dataset.

And Goldberger A. , Mosaic organization of DNA nucleotides, Phys. Rev. E Stat. Nonlin. Soft Matter Phys. , D. Schertzer, S. Lovejoy, Mulitfractal Fluctuations in Finance, International Journal of Theoretical and Applied Finance, 3, 361–364 (2000) Vandewalle N. and M. , Stanley H. , Scaling and memory in volatility return intervals in financial markets, PNAS, (102) 26, 9424–9428 (2005) Bounds for Concave Distortion Risk Measures for Sums of Risks∗ Antonella Campana and Paola Ferretti Summary. In this paper we consider the problem of studying the gap between bounds of risk measures of sums of non-independent random variables.

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