Stochastic Differential Equations and Related Topics

主要研究人员建议研究随机微分方程一般领域中的六个问题及其在金融和保险中的应用。研究了正倒向随机微分方程理论中的几个长期存在的问题，主要是在非马尔科夫框架下，允许较少的正则系数和任意持续时间。应该是``用户指南‘类型的结果。在非线性Feynman-Kac公式的基础上研究了一类拟线性倒向随机偏微分方程组(BSPDE)，并首次探索了一类由实际应用直接产生的系数具有间断的新型倒向随机偏微分方程组。提出了一类具有随机终止时间和可能部分观测的组合最优再保险投资问题，作为最新结果在随机系统上的直接应用。提出了关于部分信息信用风险模型的两个问题。一种假设是所谓的‘假设(H)’(或‘’浸没性质‘)，它着眼于一类具有超线性增长和外生跳跃的特殊BSDE，着眼于涉及可违约资产的效用优化问题；另一种则试图理解假设(H)失效的过滤模型中违约的条件密度和违约强度之间的关系。PI还建议通过首先建立一个新的极限订单账簿(LOB)动态模型来研究“订单驱动”市场中的最优执行问题，该模型使用平衡密度论证，从而导致LOB的一般类型的非线性/随机形状。拟议的研究旨在解决FBSDE和落后的SPDEs理论中的一些“最后的障碍”，以及金融和保险中的一些相关问题。关于FBSDE的拟议项目建立在PI发起的结果的基础上，并将产生一个新的解决方案来处理已悬而未决多年的案件，这反过来将有助于拟议的涉及可违约资产的最优再保险/投资和效用优化问题的研究。研究部分信息信用风险模型和最优清算问题，旨在探索随机金融中新的建模方法。研究中的大多数项目都与应用领域有直接或间接的联系，特别是随机控制和随机金融/保险。使用拟议的理论研究中开发的工具，有三个问题与金融和保险问题直接相关，而一个理论问题来自洛杉矶一家当地银行的真实项目。这项拟议研究的几个部分涉及博士生和博士后研究员。PI将继续通过由南加州大学数学金融项目赞助的定期数学金融研讨会系列来加强与当地金融界的联系，PI是该项目的主任。参与这项拟议研究的博士生更有可能从当地银行获得实习机会，其中一些人可能会获得永久就业。

The principal investigator proposes to study six problems in the general area of stochastic differential equations and their applications in finance and insurance. Several long standing problems in the theory of Forward-backward Stochastic Differential Equations (FBSDEs) are investigated, mainly under the non-Markovian framework, allowing less regular coefficients and arbitrary time duration. A ``User's Guide" type result is expected. A class of quasilinear Backward Stochastic PDEs (BSPDE) is studied in the spirit of nonlinear Feynman-Kac formula, and a new type of FBSDEs with coefficients having discontinuity, arising directly from a real application, will be explored for the first time. A class of combined optimal reinsurance and investment problems with random terminal times and possible partial observations is proposed as a direct application of the newly developed results on FBSDEs. Two problems regarding credit risk models with partial information are proposed. One assumes the so-called ``Hypothesis (H)" (or ``immersion property") and focuses on a special BSDE with super-linear growth and exogenous jumps, with an eye on the utility optimization problems involving defaultable assets; and the other tries to understand the relationship between the conditional density and intensity of the defaults in filtering models where the Hypothesis (H) fail. The PI also proposes to investigate optimal execution problems in an ``order-driven" market by first establishing a new model for the dynamics of the Limit Order Book (LOB) using an equilibrium density argument that results in a general type of nonlinear/random shape of the LOB. The proposed research aims at resolving some of the ``last obstacles" in the theory of FBSDEs and backward SPDEs, and some related topics in finance and insurance. The proposed projects on FBSDEs build on the results initiated by the PI and will lead to a new solution scheme to treat cases that have been open for many years, which will in turn help the proposed research on optimal reinsurance/investment and utility optimization problems involving defautable assets. The study on credit risk models with partial information and optimal liquidation problems are aiming at exploring new modeling aspects in stochastic finance. Most projects in the proposed research have direct or indirect connections to applied fields, especially stochastic control and stochastic finance/insurance. Three problems are directly related to issues in finance and insurance, using the tools developed in the proposed theoretical studies, while one theoretical problem arises from a real project with a local bank in LA. Several parts of the proposed research involve Ph.D students and postdoctoral fellows. The PI will continue strengthening the connections with local financial communities through a regular Math Finance Colloquium series sponsored by the Math Finance Program at USC, for which PI is the director. Ph.D students involved in the proposed research are more likely to obtain internships from the local banks, and some might lead to permanent employment.