Backward Stochastic Partial Differential Equations: Theory and Applications in Stochastic Control and Mathematical Finance

后向随机偏微分方程：随机控制和数学金融的理论与应用

• 批准号: RGPIN-2018-04325
• 负责人: Qiu, Jinniao
• 金额: $ 1.68万
• 依托单位: University of Calgary
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759105
• 关键词: Backward; Stochastic; Partial; Differential; Equations

This research proposal focuses on the theory and applications of backward stochastic partial differential equations (BSPDEs). Such BSPDEs arise naturally in many applications of probability theory and stochastic processes, especially in mathematical finance and stochastic control. For instance, in the utility maximization with random coefficients the BSPDE is raised as the stochastic Hamilton-Jacobi-Bellman (HJB) equation to characterize the value function and optimal strategies, and in the nonlinear filtering and stochastic control under incomplete information, it can be the adjoint equation of Duncan-Mortensen-Zakai filtration equation. Nevertheless, the mathematical theory of BSPDEs are far from complete. In particular, the wellposedness of fully nonlinear stochastic HJB equations was proposed by Peng in 1992 and claimed to be an open problem first in 1999 and then in his plenary lecture of ICM 2010. The applicant has proposed an ambitious schedule to study the wellposedness of fully nonlinear stochastic HJB equations and some new classes of BSPDEs and discuss their applications in stochastic control and mathematical finance. The proposal involves two long-term aims and two short-term aims.The first long-term aim is the wellposedness of fully nonlinear stochastic HJB equations. The applicant intends to establish a fairly complete theory of viscosity solutions including three main topics: the general uniqueness, the regularity estimate and the construction of optimal feedback controls for general cases.The second long-term aim is to develop the discrete approximations of BSPDEs. The applicant will start from the numerical analysis for semilinear BSPDEs on domains, then turn to numerical approximations for coupled systems of forward-backward SPDEs, and finally for the fully nonlinear stochastic HJB equations.In the first short-term aim, the applicant will study the optimal control problems of reflected stochastic differential equations and associated BSPDEs with Neumann boundary conditions. Applications include controlled queueing problems and power controls in wireless communications.The second short-term aim is devoted to the optimal liquidation in target zone models, a type of stochastic optimal control problems of stochastic differential equations with obstacles and terminal state constraints.The research program is devoted to a fairly complete theory of stochastic control and associated BSPDEs that are tailor-made to study models of optimal decision making under uncertainty, especially in the areas of energy, commodity and environmental finance. It fits very well with and complements the research activities of the applied probability groups in Canada. Advanced methods will be developed and applications will be discussed. The involved undergraduate, graduate and postdoctoral researchers will have training opportunities in relevant fields.

本研究计划重点关注后向随机偏微分方程（BSPDE）的理论和应用。这种 BSPDE 自然出现在概率论和随机过程的许多应用中，特别是在数学金融和随机控制中。例如，在随机系数效用最大化中，BSPDE被提出为随机Hamilton-Jacobi-Bellman (HJB)方程来刻画价值函数和最优策略，而在不完全信息下的非线性滤波和随机控制中，它可以是Duncan-Mortensen-Zakai滤波方程的伴随方程。然而，BSPDE 的数学理论还远未完善。特别是，完全非线性随机 HJB 方程的适定性由 Peng 于 1992 年提出，并在 1999 年和他在 ICM 2010 的全体演讲中首次声称是一个开放问题。申请人提出了一个雄心勃勃的计划来研究完全非线性随机 HJB 方程和一些新类 BSPDE 的适定性，并讨论它们在随机控制和数学金融中的应用。该提案涉及两个长期目标和两个短期目标。第一个长期目标是完全非线性随机HJB方程的适定性。申请人打算建立一个相当完整的粘度解理论，包括三个主要主题：一般唯一性、规律性估计和一般情况下最优反馈控制的构建。第二个长期目标是开发 BSPDE 的离散近似。申请人将从域上半线性 BSPDE 的数值分析开始，然后转向前向-后向 SPDE 耦合系统的数值近似，最后是完全非线性随机 HJB 方程。在第一个短期目标中，申请人将研究反射随机微分方程和具有诺伊曼边界条件的相关 BSPDE 的最优控制问题。应用包括无线通信中的受控排队问题和功率控制。第二个短期目标致力于目标区域模型中的最优清算，这是一种带有障碍和终端状态约束的随机微分方程的随机最优控制问题。该研究计划致力于相当完整的随机控制理论和相关的 BSPDE，这些理论是为研究不确定性下的最优决策模型而定制的，特别是在以下领域： 能源、商品和环境金融。它非常适合并补充加拿大应用概率组的研究活动。将开发先进方法并讨论应用。参与的本科生、研究生和博士后研究人员将获得相关领域的培训机会。