非线性随机瓮模型的随机逼近观点
Nonlinear Randomized Urn Models: a Stochastic Approximation Viewpoint
摘要：
本文扩展了随机逼近（SA）理论和Laruelle，Pag{è}s（2013）中提出的随机瓮模型之间的联系，以及它们在Bai，HU（1999，2005）和Bai，Hu，Shen（2002）中介绍的临床试验中的应用。我们不再假设绘制规则在瓮的球之间是均匀的（包含d种颜色），但可以通过函数f来加强。这是一种模拟风险规避的方法。首先，考虑f是凹的或凸的，并通过将瓮组合的动力学重新表述为带余数的SA算法，我们导出了a.s.收敛性和渐近正态性（中心极限定理，CLT）的正规化程序，通过调用所谓的ODE和ESTA方法。深入分析d=2的情形，可以发现两种不同的行为：当f是凹的时，有一个单一的平衡点;当f是凸的时，有一个从单一吸引平衡点到两个吸引和一个排斥平衡点的过渡阶段。最后一个设置使用关于朝向噪声和无噪声“陷阱”的非收敛的结果来解决，以便推导a.s.向其中一个吸引点收敛。其次，分析了Polya瓮的特殊情况（加法规则为单位矩阵时），仍然使用SA理论关于“陷阱”的结果。最后，这些结果被应用到一个函数的定期变化和金融中的最优资产配置。
英文摘要：
This paper extends the link between stochastic approximation (SA) theory and randomized urn models developed in Laruelle, Pag{è}s (2013), and their applications to clinical trials introduced in Bai, HU (1999,2005) and Bai, Hu, Shen (2002). We no longer assume that the drawing rule is uniform among the balls of the urn (which contains d colors), but can be reinforced by a function f. This is a way to model risk aversion. Firstly, by considering that f is concave or convex and by reformulating the dynamics of the urn composition as an SA algorithm with remainder, we derive the a.s. convergence and the asymptotic normality (Central Limit Theorem, CLT) of the normalized procedure by calling upon the so-called ODE and SDE methods. An in-depth analysis of the case d=2 exhibits two different behaviors: A single equilibrium point when f is concave, and when f is convex, a transition phase from a single attracting equilibrium to a system with two attracting and one repulsive equilibrium points. The last setting is solved using results on non-convergence toward noisy and noiseless "traps" in order to deduce the a.s. convergence toward one of the attracting points. Secondly, the special case of a Polya urn (when the addition rule is the identity matrix) is analyzed, still using result from SA theory about "traps". Finally, these results are applied to a function with regular variation and to an optimal asset allocation in Finance.
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