Boltzmann运动论在经济物理学研究中的应用

发布时间：2018-04-20 07:27

本文选题：Boltzmann方程 + 温和解　；参考：《清华大学》2012年博士论文

【摘要】：Boltzmann方程是气体运动论中的重要模型。近年来，应用Boltzmann方程的理论和方法来研究经济学的问题已经成为经济物理学的热点之一。鉴于市场中的交易者通常以投资组合的方式来降低整体风险，因此，研究投资组合的概率分布是一个非常有意义的课题。本文首先给出了一个在非负债背景下的高维模型，来描述投资组合的概率密度函数随时间的发展变化。由于本文模型中的“碰撞核”结构较为复杂且缺少对称性，因此很难直接将解写成wild sums的形式，于是我们应用压缩映像原理来证明该模型温和解的存在性和唯一性，并随之给出了该解的矩估计。本文的重点在于研究解的长时行为，因为这将有助于我们了解市场中投资组合的发展趋势。由于该模型碰撞核的结构特点，导致其温和解仅保持零阶矩守恒，即该模型中仅交易者的总数守恒。因此，即使在弱收敛的意义下，也很难直接得到非平凡的稳态解。于是，我们考虑时间尺度变换解来讨论本文模型的渐近极限方程。本文的主要结果定理5.3表明：在一定的假设条件下，当交易额较小时，该模型在长时极限的意义下，可以用高维Fokker-Planck方程来逼近。该结果较好的体现了本文高维模型的研究意义。这是因为我们所得到的高维模型的渐近极限方程并不是若干一维情形的简单叠加，而是依赖于两两投资项目之间的关系。该定理的证明过程主要包含了以下两方面的工作：一方面，我们给出了一族尺度变换解{gr(w,t)}r,t的L1弱紧性的证明。我们首先得到了解的L2估计，结合已有的解的矩估计及零阶距守恒律，应用Dunford-Pettis准则得到了该部分的结果。这部分的结论非常重要，因为如果缺少{gr(w,t)}1r,t的L弱紧性，我们所得到的渐近极限方程仅在分布意义下成立。另一方面，，与一维情形相比，我们采用了划分积分区域、将被积函数变形等方法，对两个余项1(r)、2(r)进行了更为细致的估计，从而弱化了这两个余项收敛于零的条件。
[Abstract]:Boltzmann equation is an important model in the theory of gas motion. In recent years, using the theory and method of Boltzmann equation to study economics has become one of the hotspots in economic physics. In view of the fact that traders in the market usually reduce the overall risk by portfolio, studying the probability distribution of portfolio is a very meaningful topic. In this paper, we first give a high-dimensional model under the non-debt background to describe the probability density function of the portfolio over time. Because the structure of "collision nucleus" in this model is complex and lacks symmetry, it is difficult to write the solution directly into the form of wild sums, so we apply the principle of compression mapping to prove the existence and uniqueness of the model. Then the moment estimation of the solution is given. The focus of this paper is to study the long-term behavior of solutions, because this will help us to understand the trend of portfolio development in the market. Because of the structural characteristics of the collision nucleus, the mild solution of the model only keeps the zero-order moment conservation, that is, only the total number of traders in the model is conserved. Therefore, even in the sense of weak convergence, it is difficult to obtain nontrivial steady-state solutions directly. Therefore, we consider the solution of time scale transformation to discuss the asymptotic limit equation of the model. The main result of this paper, theorem 5.3, shows that under certain assumptions, the model can be approximated by high-dimensional Fokker-Planck equation in the sense of long term limit when the transaction volume is small. The results well reflect the significance of the study of the high-dimensional model in this paper. This is because the asymptotic limit equation of the high dimensional model is not a simple superposition of some one-dimensional cases, but depends on the relationship between pairwise investment projects. The proof of the theorem mainly includes the following two aspects: on the one hand, we give a proof of the L1 weakly compactness of a family of scale transformation solutions ({gri _ w _ t)} ~ r ~ (t). First, we obtain the L2 estimate of the solution. Combining with the moment estimation of the solution and the zero-order distance conservation law, we apply the Dunford-Pettis criterion to obtain the results of this part. This part of the conclusion is very important, because if we lack the L weakly compactness of {grg ~ (w) ~ (t)} 1rt, the asymptotic limit equation we have obtained is only valid in the sense of distribution. On the other hand, compared with the one-dimensional case, we use the method of dividing the integral region and deforming the integrable function to estimate the two remainder terms 1 ~ 1 ~ 2 ~ 2 ~ r in more detail, thus weakening the condition that the two remainder terms converge to zero.
【学位授予单位】：清华大学
【学位级别】：博士
【学位授予年份】：2012
【分类号】：O211.3;F830.59

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