Stochastic Differential Equations and Applications

主要研究者建议研究随机微分方程及其在金融中的应用的一般领域的五个问题。基于一种新型的路径随机Taylor展开，提出了一种新的随机粘性解理论框架，该框架实质性地推进了完全非线性随机偏微分方程的随机粘性解理论.本文将进一步研究正倒向鞅问题（FBMP）的新概念和正倒向随机微分方程的弱解，并将其适定性，特别是解的唯一性的讨论推广到系数可测和/或VMO（Variation Mean Oscillation）的一般情形，达到理论的最高阶段.本文提出了一种新的反射倒向随机微分方程的变形，并着眼于它在金融中各种问题中的应用，其中Skorohod问题的变形和可选过程的随机表示定理是起源。有两个拟议的问题与财政关系更为密切。将一般的凸风险测度纳入新发展的滤过一致非线性期望理论的框架中，并利用二次倒向随机微分方程和BMO鞅（Bounded Mean Oscillation）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，提出了一种部分信息信用风险模型.新类型的非线性滤波问题的出现，以及一些有趣的新现象的出现，都给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领域以及金融等相关领域取得重大进展。关于非线性SPDE的随机粘性解和FBSDEs的弱解的拟议项目将建立在PI发起的结果的基础上，以进一步探索各自学科的性质，并填补长期存在的理论空白。关于变量反映的贝叶斯、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。 在拟议的研究中，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直接处理，使用随机分析和随机微分方程的先进技术。部分研究涉及博士生和博士后研究员，部分反映了这一建议的教育激励。

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.