Stochastic Differential Equations and Applications

主要研究随机微分方程一般领域中的五个问题及其在金融中的应用。提出了一种基于新型路径随机泰勒展开的一般框架，从而极大地推进了长期存在的全非线性随机偏微分方程的随机粘度解理论。本文将进一步研究前向倒向鞅问题（FBMP）的新概念和前向倒向SDEs的弱解，并将其适定性，特别是解的唯一性的讨论扩展到允许系数可测量和/或VMO（变均振荡）的一般情况，达到理论的最先进阶段。本文提出了反射后向SDEs的一种新变体，并将其应用于金融中的各种问题，其中提出了Skorohod问题的一个变体和可选p过程的随机表示定理。提出的两个问题与金融更密切相关。一般凸风险测度将被纳入新发展的滤波一致非线性期望理论的框架，并将使用二次倒向随机微分方程和BMO（有界平均振荡）鞅理论进行研究。针对连续观测、计数过程观测和延迟信息同时存在的一般框架，提出了部分信息信用风险模型。新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理论提出了新的问题。本研究将在随机微分方程及相关领域如金融等方面寻求重大进展。拟开展的非线性SPDEs的随机粘度解和FBSDEs的弱解项目将以PI的成果为基础，进一步探索各自学科的性质，并填补长期存在的理论空白。关于变异反映BSDE、二次非线性期望和部分信息信用风险模型的项目旨在开发随机分析的新工具，以解决金融中复杂但实际的问题。拟研究的项目大多与应用领域有直接或间接的联系，尤其是与随机控制、随机金融、运筹学等领域。在金融理论中的两个问题将直接处理，使用先进的技术在随机分析和随机微分方程。拟议的研究有几个部分涉及博士生和博士后，这在一定程度上反映了该提案的教育激励。

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.