Logarithme naturel

Résumé

Le logarithme naturel ou logarithme népérien, ou encore logarithme hyperbolique jusqu'au , transforme, comme les autres fonctions logarithmes, les produits en sommes. L'utilisation de telles fonctions permet de faciliter les calculs comprenant de nombreuses multiplications, divisions et élévations à des puissances rationnelles. Il est souvent noté ln(). Le logarithme naturel ou népérien est dit de base e car ln(e) = 1. Le logarithme népérien d'un nombre x peut également être défini comme la puissance à laquelle il faut élever e pour obtenir x. La fonction logarithme népérien est donc la bijection réciproque de la fonction exponentielle. C'est également la primitive définie sur les réels strictement positifs et qui s'annule en 1 de la fonction inverse x ↦ 1/''x''. Cette fonction fut notée l. ou l, dès le début du , et jusque dans la première moitié du , puis log. ou log dès la fin du , puis Log pour la différencier de la fonction log (logarithme de base quelconque, ou

Source officielle

https://fr.wikipedia.org/wiki/Logarithme_naturel

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The Hausdorff measure of the range and level sets of Gaussian random fields with sectorial local nondeterminism

Cheuk Yin Lee

We determine the exact Hausdorff measure functions for the range and level sets of a class of Gaussian random fields satisfying sectorial local nondeterminism and other assumptions. We also establish a Chung-type law of the iterated logarithm. The results can be applied to the Brownian sheet, fractional Brownian sheets whose Hurst indices are the same in all directions, and systems of linear stochastic wave equations in one spatial dimension driven by space-time white noise or colored noise.

INT STATISTICAL INST2022

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Dating Death: An Empirical Comparison of Medical Underwriters in the U.S. Life Settlements Market

Jiahua Xu

The value of a life settlement investment, manifested through a traded life insurance policy, is highly dependent on the insured's life expectancy (LE). LE estimation in life settlements relies heavily on medical underwriting. Employing different evaluation processes, underwriters rarely agree on LE estimates, leading to valuation disparities. We use the natural logarithm of the implied mortality multiplier (ln k) to compare the underwriting results of the four major U.S. medical underwriters (ITM, AVS, Fasano, and LSI). ln k is normalized in terms of gender, age, and smoking status, and is therefore a more suitable indicator than LE estimates for highlevel comparisons, especially when the compared groups have a heterogeneous make-up. Based on the analysis of life settlement samples from 2011 to 2016, we trace patterns of underwriters' ln k in both secondary and tertiary markets of life settlements, and investigate systematic differences in their estimation. Our results show that an underwriter can, relative to peers, act more conservatively (issuing longer LE estimates) for one cohort while more aggressively (issuing shorter LE estimates) for another. We also detect signs of intermediaries' cherry-picking behavior and discuss additional theories that shed light on the convoluted LE landscape.

2020

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Let X = {X(t); t ∈ RN} be a (N,d) fractional Brownian motion in Rd of index H ∈ (0,1). We study the local time of X for all temporal dimensions N and spatial dimensions d for which local time exist. We obtain two main results : R1. If we denote by Lx(I) the local time of X at x on I ⊂ RN, then there exists a positive finite constant c such that mφ(X-1(0) ∩ [0,1]N) = c L0([0,1]N), where φ(r) = rN-dH (log log 1/r)dH/N and mφ(E) is the Hausdorff φ-measure of E. This solves the problem of the relationship between the local time and the exact Hausdorff measure of zero set for X. R2. We refine results of Xiao (1997) for the local times of (N,d) fractional Brownian motion. We prove the law of iterated logarithm and global Hölder condition for the local time for our process. These results establish interesting properties which were only partially proved in the literature. This literature began with the work of Taylor and Wendel (1966) and Perkins (1981) for the first result; Kesten (1965) and Perkins (1981) for the second on the Brownian motion. It continued with several works of which that of Xiao (1997) on the locally nondeterministic processes with stationnary increments including the (N,d) fractional Brownian motion. An intermediate result is found to solve the case N > 1. We generalize the result of Kasahara et al. (1999) on the tail probability of local time.

EPFL2008

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