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Title: Parabolic systems and stochastic controls: nonlocality, nonlinearity, and time-inconsistency
Authors: Lei, Qian
Keywords: Science::Mathematics::Probability theory
Science::Mathematics::Applied mathematics::Game theory
Science::Mathematics::Applied mathematics::Optimization
Business::Finance::Mathematical finance
Issue Date: 2022
Publisher: Nanyang Technological University
Source: Lei, Q. (2022). Parabolic systems and stochastic controls: nonlocality, nonlinearity, and time-inconsistency. Doctoral thesis, Nanyang Technological University, Singapore.
Abstract: This thesis aims to advance the theories of partial differential equation (PDE) and stochastic differential equation (SDE), and by which, we address decade-long open problems in the field of stochastic controls. We develop systematically a theory of nonlocal parabolic systems in aspects of existence, uniqueness, stability, and computational method, where there is an external time parameter t on top of the temporal and spatial variables (s, y). The nonlocality comes from the two time variable structure. Such equations arise from time-inconsistent problems in game theory or behavioral economics, where the observations and preferences are (reference-)time-dependent. This thesis first obtains the well-posedness of nonlocal linear systems and establishes a Schauder-type prior estimate for the solutions with an innovative construction of appropriate norms and Banach spaces and contraction mappings over which. Subsequently, we take advantage of linearization methods and quasilinearization methods to establish the well-posedness results of solutions under the semilinear, quasilinear, and fully nonlinear case. Besides of pushing the frontiers of PDE, our theoretical framework allows the control variate entering the diffusion of state process, which breaks successfully through the existing bottleneck of time-inconsistent stochastic control problems. Moreover, we also provide a general and unified treatment for the Feynman-Kac formulas of a flow of forward-backward SDEs.
DOI: 10.32657/10356/161078
Schools: School of Physical and Mathematical Sciences 
Rights: This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0).
Fulltext Permission: open
Fulltext Availability: With Fulltext
Appears in Collections:SPMS Theses

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