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

本课题主要研究倒向随机偏微分方程的理论与应用。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，自然会出现这种bspde。例如，在具有随机系数的效用最大化问题中，BSPDE被提出为表征值函数和最优策略的随机Hamilton-Jacobi-Bellman （HJB）方程；在不完全信息下的非线性滤波和随机控制问题中，它可以被提出为Duncan-Mortensen-Zakai滤波方程的伴随方程。然而，BSPDEs的数学理论还远远不够完善。特别是，彭在1992年提出了完全非线性随机HJB方程的完备性，并在1999年首次声称是一个开放问题，然后在2010年ICM的全体演讲中。申请人提出了一个雄心勃勃的计划，研究完全非线性随机HJB方程和一些新的BSPDEs类的适定性，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，正则性估计和一般情况下最优反馈控制的构建。第二个长期目标是发展BSPDEs的离散逼近。申请者将从域上的半线性BSPDEs的数值分析开始，然后转向前向向后SPDEs耦合系统的数值逼近，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的最优控制问题以及具有诺伊曼边界条件的相关BSPDEs。应用包括控制排队问题和无线通信中的电源控制。第二个短期目标是研究目标区域模型的最优清算问题，这是一类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。该研究项目致力于一个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量身定制的，特别是在能源、商品和环境金融领域。它与加拿大应用概率组的研究活动非常吻合并互为补充。将开发先进的方法并讨论其应用。参与的本科生、研究生和博士后研究人员将有相关领域的培训机会。

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.