经济学和金融学中基于个体的模型的极限定理
Limit Theorems for Individual-Based Models in Economics and Finance
摘要：
最近，人们对使用统计物理学的思想来模拟经济学和金融学中的某些类型的问题产生了广泛的兴趣。其主要思想是从代理人之间的随机局部相互作用中推导出市场的宏观行为。我们的目的是提出一个一般的框架，包括广泛的模型，通过证明一个法律的大数定律和中心极限定理的某些相互作用的粒子系统非常一般的状态空间。为了做到这一点，我们从数学生态学和数学物理的一些工作中得到启发。第一个结果是证明了系统视为一个测度值的过程，而要证明第二个，我们将需要引入一个链的嵌入一些抽象的Banach和希尔伯特空间的检验函数，并证明波动收敛到一定的广义高斯随机微分方程的解决方案采取的对偶空间之一。
英文摘要：
There is a widespread recent interest in using ideas from statistical physics to model certain types of problems in economics and finance. The main idea is to derive the macroscopic behavior of the market from the random local interactions between agents. Our purpose is to present a general framework that encompasses a broad range of models, by proving a law of large numbers and a central limit theorem for certain interacting particle systems with very general state spaces. To do this we draw inspiration from some work done in mathematical ecology and mathematical physics. The first result is proved for the system seen as a measure-valued process, while to prove the second one we will need to introduce a chain of embeddings of some abstract Banach and Hilbert spaces of test functions and prove that the fluctuations converge to the solution of a certain generalized Gaussian stochastic differential equation taking values in the dual of one of these spaces.
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