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

本文主要研究倒向随机偏微分方程(BSPDEs)的理论和应用。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，这种BSPDEs是自然而然出现的。例如，在具有随机系数的效用最大化中，BSPDE可以作为随机Hamilton-Jacobi-Bellman(HJB)方程来刻画价值函数和最优策略；在不完全信息下的非线性滤波和随机控制中，BSPDE可以作为Duncan-Mortensen-Zakai渗流方程的伴随方程。然而，BSPDEs的数学理论还远远不完整。特别是，完全非线性随机HJB方程的适定性是由彭在1992年提出的，并在1999年和2010年的ICM全体会议上声称是一个公开的问题。 申请人提出了一个雄心勃勃的时间表，研究完全非线性随机HJB方程和一些新的BSPDEs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比较完整的粘性解理论，包括三个主要主题：一般情况下的一般唯一性、正则性估计和最优反馈控制的构造。 第二个长期目标是开发BSPDEs的离散近似。申请者将从区域上半线性BSPDEs的数值分析开始，然后转向正反向SPDEs耦合系统的数值逼近，最后转向完全非线性随机HJB方程的数值逼近。 在第一个短期目标中，申请者将研究反射随机微分方程的最优控制问题，以及与Neumann边界条件相关的BSPDEs。应用包括无线通信中的受控排队问题和电源控制。 第二个短期目标致力于目标区模型的最优清算，这是一类带有障碍和终端状态约束的随机微分方程组的随机最优控制问题。 该研究项目致力于研究相当完整的随机控制理论和相关的BSPDEs，这些理论是为研究不确定条件下的最优决策模型而量身定做的，特别是在能源、大宗商品和环境金融领域。它很好地契合和补充了加拿大应用概率小组的研究活动。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获得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.