Some Topics on Path Dependent Partial Differential Equations and Stochastic Differential Equations

在许多实际问题中，所研究的对象可以被刻画为某些偏微分方程的唯一解。 这就要求问题具有所谓的马尔可夫结构，即对象只依赖于某个底层过程的当前值。 在许多应用中，例如在许多金融模型中，问题可能是非马尔可夫的，也就是说，对象依赖于底层过程的整个历史。 研究者和他的同事们研究所谓的路径依赖偏微分方程，这将使他们能够将标准的思想和技术应用于更大类的应用问题。 然而，主要的困难在于，在这样一个非马尔可夫框架中，对象过程通常是不光滑的，因此不能被理解为经典意义上的方程的解。 这个项目的主要目的是研究这类方程的某些弱解，称为粘性解。 研究人员认为，确保这些解决方案的存在性，唯一性和稳定性的条件，并开发数值方法来计算它们。 他还通过随机微分方程研究时间不一致问题。 在前景理论和契约理论等许多经济和金融应用中，这些问题自然会出现。 研究生将在项目过程中接受培训。路径相关偏微分方程理论将路径视为变量，著名的例子包括路径相关的哈密尔顿-雅可比-贝尔曼方程和艾萨克方程。 它为具有扩散控制的随机优化问题以及非马尔科夫环境下具有波动不确定性的经济或金融模型提供了一个方便的框架。 研究人员和他的同事们以前已经开发了一个理论的粘性解决方案的问题，其中的状态空间不是局部紧凑和主导收敛定理失败。 在这个项目中，他在三个方向上进一步发展了理论，主要是出于各种应用的动机：（i）将偏微分方程标准粘性理论中使用的凸分析扩展到路径依赖设置，并在较弱的条件下为比较原理提供新的证明;（ii）调查粘性解的正则性;（iii）提出可行的数值方法。 研究者还考虑了时间不一致的问题。 通常有两种类型的策略来解决这些问题：预先承诺和一致的计划。 大多数连续时间模型的研究集中在后者。 提出了一种新的方法，利用多维受控倒向随机微分方程求解前者。 本文的主要创新之处在于将时间不一致性与比较原则联系起来，并引入了一种新的比较顺序。 该方法也可能为一些多维倒向随机微分方程的公开问题，其中缺乏一个标准的比较原则提供新的见解。

In many practical problems the object under study can be characterized as the unique solution to certain partial differential equations. This requires the so-called Markov structure of the problem, namely, the object depends only on the current value of a certain underlying process. In many applications, for example in many financial models, the problem may be non-Markovian, namely, the object depends on the whole history of the underlying process. The investigator and his colleagues study what are called path-dependent partial differential equations, which would enable them to apply standard ideas and techniques to a much larger class of applied problems. The major difficulty, however, is that in such a non-Markovian framework the object process is typically not smooth and thus cannot be understood as the solution to the equation in a classical sense. The main aim of this project is to study certain weak solutions, called viscosity solutions, for such equations. The investigator considers conditions that ensure the existence, uniqueness, and stability of these solutions, and develops numerical methods to compute them. He also studies time-inconsistent problems via stochastic differential equations. Such problems arise naturally in many economic and financial applications such as prospect theory and contract theory. Graduate students will be trained in the course of the project. The theory of path-dependent partial differential equations considers paths to be a variable; notable examples including path-dependent Hamilton-Jacobi-Bellman equations and Isaacs equations. It provides a convenient framework for stochastic optimization problems with diffusion control and economic or financial models with volatility uncertainty in a non-Markovian setting. The investigator and his colleagues have previously developed a theory of viscosity solutions for problems in which the state space is not locally compact and the dominated convergence theorem fails. In this project he develops the theory further in three directions, mainly motivated by various applications: (i) extending the convex analysis used in standard viscosity theory for partial differential equations to the path-dependent setting and providing a new proof for the comparison principle under weaker conditions; (ii) investigating the regularity of viscosity solutions; and (iii) proposing feasible numerical methods. The investigator also considers time-inconsistent problems. There are typically two types of strategies for such problems: pre-commitment and consistent planning. Most studies of continuous time models focus on the latter one. A novel approach for the former one is proposed, by using multidimensional controlled backward stochastic differential equations. The main innovation is to relate the time inconsistency to a comparison principle and to introduce a new order for comparison. The approach may also provide new insights for some open problems on multidimensional backward stochastic differential equations, which lack a standard comparison principle.