Study on the nonlinear partial differential equations arising in applied field

本文对应用领域中出现的非线性偏微分方程进行了研究，目的是深入理解偏微分方程所代表的现象。考虑最优投资问题中的Hamilton-Jacobi-Bellman方程。我们导出了略一般的非线性风险偏好偏微分方程，并分析了方程。此外，我们还研究了描述随机变量间非线性关系的联结函数。我们介绍了copula的时间演化。这两个结果都值得推动对经济学现象的理解。

The study is performed on nonlinear partial differential equations (PDEs) arising in applied field with the purpose of deeply understanding the phenomena represented by the PDEs. We consider the Hamilton-Jacobi-Bellman equation in the optimal investment problem. We derive the slightly general nonlinear PDEs for the risk preference and analyze the equation. Moreover, we study the copula function, which describes the nonlinear relation among random variables. We introduce the time evolution of copulas.Both results are worth advancing the understandings of economics phenomena.