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

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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.