相对与离散效用最大化熵

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摘要翻译：
由Slomczynski和Zastawniak（Ann.Prob 32（2004）2261-2285,arxiv:Math.PR/0410115 v1)引入并研究的概率密度效用最大化熵（U-熵）的概念在两个方向上得到了扩展。首先定义了任意概率空间中两个概率测度的相对U熵。然后，针对离散概率空间，引入了概率测度的绝对U熵。这两个概念都基于从数学金融学中借来的想法，即最大化投资者最终财富的预期效用。此外，U熵在热力学中也是相关的，因为它可以取代第二定律中的标准玻尔兹曼-香农熵。如果效用函数是对数或等弹性（幂函数），则恢复了众所周知的Boltzmann-Shannon和Renyi相对熵的概念。我们建立了相对U熵和离散U熵的主要性质，并讨论了它们与文献中几种相关方法的联系。
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英文标题：
《Relative and Discrete Utility Maximising Entropy》
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作者：
Grzegorz Hara\'nczyk, Wojciech S{\l}omczy\'nski, Tomasz Zastawniak
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Quantitative Finance        数量金融学
二级分类：Statistical Finance        统计金融
分类描述：Statistical, econometric and econophysics analyses with applications to financial markets and economic data
统计、计量经济学和经济物理学分析及其在金融市场和经济数据中的应用
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英文摘要：
  The notion of utility maximising entropy (u-entropy) of a probability density, which was introduced and studied by Slomczynski and Zastawniak (Ann. Prob 32 (2004) 2261-2285, arXiv:math.PR/0410115 v1), is extended in two directions. First, the relative u-entropy of two probability measures in arbitrary probability spaces is defined. Then, specialising to discrete probability spaces, we also introduce the absolute u-entropy of a probability measure. Both notions are based on the idea, borrowed from mathematical finance, of maximising the expected utility of the terminal wealth of an investor. Moreover, u-entropy is also relevant in thermodynamics, as it can replace the standard Boltzmann-Shannon entropy in the Second Law. If the utility function is logarithmic or isoelastic (a power function), then the well-known notions of the Boltzmann-Shannon and Renyi relative entropy are recovered. We establish the principal properties of relative and discrete u-entropy and discuss the links with several related approaches in the literature.
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PDF链接：
https://arxiv.org/pdf/0709.1281

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关键词：效用最大化 最大化 Applications Differential Mathematical 投资者 对数 Prob 离散