Stochastic Differential Equations and Applications

主要研究人员建议研究随机微分方程及其在金融中的应用的一般领域中的五个问题。提出了一种基于新型路径随机泰勒展开式的一般框架，大大推进了长期存在的完全非线性随机偏微分方程解的随机粘性理论。我们将进一步研究前向后向鞅问题(FBMP)的新概念和前向后向随机微分方程的弱解，并将适定性的讨论，特别是解的唯一性的讨论，推广到一般情形，其中系数是可测的和/或变分平均振荡(VMO)，达到理论的最高级阶段。提出了一种新的反射倒向随机微分方程，并讨论了它在各种金融问题中的应用，其中提出了一种Skorohod问题的变体和一个可选过程的随机表示定理。提出的两个问题与金融更密切相关。一般的凸风险度量将被置于新发展的过滤相容非线性期望理论的框架内，并将使用二次倒向随机微分方程和BMO(有界平均振荡)鞅理论来研究。针对同时存在连续观测、计数过程观测和延迟信息的一般框架，提出了一个部分信息信用风险模型。新类型的非线性滤波问题有望出现，一些有趣的新现象也为随机微分方程论提出了新的问题。所提出的关于非线性SPDEs的随机粘性解和FBSDE的弱解的项目将建立在PI所开创的结果的基础上，以进一步探索各自学科的性质，并填补长期存在的理论的空白。变量反映BSDE、二次非线性期望和部分信息信用风险模型的研究旨在开发新的随机分析工具，以解决金融中复杂而又实际的问题。建议研究中的大多数项目都与应用领域有直接或间接的联系，特别是随机控制、随机金融和运筹学。金融理论中的两个问题将被直接处理，使用随机分析和随机微分方程中的先进技术。这项拟议研究的几个部分涉及博士生和博士后研究员，部分反映了这一提议的教育动机。

The principal investigator proposes to study five problems in the general area of stochastic differential equations and their applications in finance. A general framework based on a new type of pathwise stochastic Taylor expansion is proposed to substantially advance the long standing theory of stochastic viscosity solution for fully nonlinear stochastic partial differential equations. The new notion of forward-backward martingale problem (FBMP) and the weak solution to forward-backward SDEs will be further investigated, and the discussion of well-posedness, especially the uniqueness of the solutions will be extended to general cases where the coefficients are allowed to be measurable and/or VMO (Variation Mean Oscillation), reaching the most advanced stage of the theory. A new variant of reflected backward SDEs is proposed with an eye on its applications to various problems in finance where a variant of Skorohod problem and a stochastic representation theorem for optional p rocesses were originated. Two proposed problems are more closely related to finance. The general convex risk measures will be put into the framework of the newly developed theory of filtration consistent nonlinear expectations, and will be investigated using quadratic backward stochastic differential equations and the BMO (Bounded Mean Oscillation) martingale theory. A credit risk model with partial information is proposed, aiming at a general framework where continuous observation, counting process observation, and delayed information can be present at the same time. New types of nonlinear filtering problems are expected to emerge, and some interesting new phenomena exhibited so far have raised new questions for the theory of stochastic differential equations.The proposed research is seeking significant advancement in the field of stochastic differential equations, as well as the related areas such as finance. The proposed projects on stochastic viscosity solution for nonlinear SPDEs and weak solution of FBSDEs will build on the results initiated by the PI to further explore the nature of the respective subjects, and to fill the gaps in the long standing theory. The projects on variant reflected BSDE, on quadratic nonlinear expectations, and on credit risk models with partial information are aiming at developing new tools in stochastic analysis to solve complex but practical problems in finance. Most projects in the proposed research have direct or indirect connections to applied fields, especially stochastic control, stochastic finance, and operations research. Two problems in finance theory will be treated directly, using advanced techniques in stochastic analysis and stochastic differential equations. Several parts of the pro posed research involve Ph.D students and postdoctoral fellows, partly reflecting an educational incentive of this proposal.