价格形成中一个自由边界模型的渐近性
Asymptotics for a free-boundary model in price formation
摘要：
本文研究了具有非标准测度值右端的一维抛物型发展方程解的大时间渐近性，其中包含解在自由边界点处的导数.该问题是一个特殊的情况下，平均场自由边界模型提出的Lasry-Lions的价格形成和动态均衡。证明的主要步骤是基于线性化问题中自由边界消失的事实，从而可以通过半群理论作为扰动来处理。这需要对函数空间进行精细的选择，因为在自由边界附近需要更高的正则性。我们证明了全局解的存在性，初始数据在任何平衡点的一个小的邻域，并指数衰减到一个平稳的状态。此外，该方程的平衡族是稳定的，如下从中心流形理论。
英文摘要：
We study the asymptotics for large time of solutions to a one dimensional parabolic evolution equation with non-standard measure-valued right hand side, that involves derivatives of the solution computed at a free boundary point. The problem is a particular case of a mean-field free boundary model proposed by Lasry-Lions on price formation and dynamic equilibria. The main step in the proof is based on the fact that the free boundary disappears in the linearized problem, thus can be treated as a perturbation through semigroup theory. This requires a delicate choice for the function spaces since higher regularity is needed near the free boundary. We show global existence for solutions with initial data in a small neighborhood of any equilibrium point, and exponential decay towards a stationary state. Moreover, the family of equilibria of the equation is stable, as follows from center manifold theory.
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