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

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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.