Time-inconsistent control problems, path-dependent PDEs, and neural network approximation

• Main Author: Nguwi, Jiang Yu
• Other Authors: Nicolas Privault
• Format: Thesis-Doctor of Philosophy
• Language: English
• Published: Nanyang Technological University 2022
• Subjects: Science::Mathematics
• Online Access: https://hdl.handle.net/10356/163165
• Institution: Nanyang Technological University

There are two main parts of this thesis: Time-Inconsistent Control (TIC) problems (Chapters 1 and 2) and the neural network approximation (Chapters 3 and 4). In the first part of the thesis, we are interested in constructing and characterizing the equilibria of TIC problems, while in the second part of the thesis, we apply the neural network approach to solve path-dependent Partial Differential Equations (PDEs) and derive the generalization error bounds of 2-layer neural networks. More precisely, in Chapter 1 we extend the construction of equilibria for mean-variance portfolio and linear-quadratic problems available in the literature to a large class of mean-field continuous-time TIC problems. Our approach relies on a time discretization of the control problem using n-person games characterized by Backward Stochastic Differential Equations (BSDEs). The existence of equilibria is proved by applying weak convergence arguments to the n-person games solutions. Then, we present the numerical results by approximating n-person games with finite Markov chains. We also show the small-time uniqueness of equilibria under Lipschitz assumptions on the Hamiltonian function. In Chapter 2, we derive a characterization of equilibria using the Malliavin calculus. For this, we replace the classical duality analysis of adjoint BSDEs by the Malliavin integration by parts. This results in a necessary and sufficient Malliavin maximum principle, which is more explicit than classical maximum principle. We apply the results to the linear-quadratic TIC problem and the generalized Merton problem. In Chapter 3, we use the neural network function to approximate path-dependent PDE solutions, and derive the error bounds of the approximation scheme. In the literature, path-dependent PDEs are solved by estimating the conditional expectations of their probabilistic representation using the regression. However, the regression approach requires a careful selection of the functional basis, which may not be possible in many cases. Our neural network approach, on the other hand, does not require such basis selection. In our numerical examples of two-person zero-sum game, Asian and barrier option pricing, our algorithm appears to be more accurate, especially in large dimensions. In Chapter 4, we derive the generalization error bound of a 2-layer neural network trained by the stochastic gradient method. Our bounds require no boundedness conditions on the loss function or on its gradients, and can be computed prior to the training of the model. In the numerical simulations, we confirm our dimension-free bound in the case of independent test data.