arXiv subjects

Hang Cheung

Publications and source records attributed to Hang Cheung.

4 recordsLinked to original sources

arXiv2025

Viscosity Solutions of Fully second-order HJB Equations in the Wasserstein Space

In this paper, we show that the value functions of mean field control problems with common noise are the unique viscosity solutions to fully second-order Hamilton-Jacobi-Bellman equations, in a Crandall-Lions-like framework. We allow the second-order derivative in measure to be state-dependent and thus infinite-dimensional, rather than derived from a finite-dimensional operator, hence the term ''fully''. Our argument leverages the construction of smooth approximations from particle systems developed by Cosso, Gozzi, Kharroubi, Pham, and Rosestolato [Trans. Amer. Math. Soc., 2023], and the compactness argument via penalization of measure moments in Soner and Yan [Appl. Math. Optim., 2024]. Our work addresses unbounded dynamics and state-dependent common noise volatility, and to our knowledge, this is the first result of its kind in the literature.

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arXiv2024

Feynman-Kac Formula for Time-Dependent Nonlinear Schr\"odinger Equations with Applications in Numerical Approximations

In this paper, we present a novel Feynman-Kac formula and investigate learning-based methods for approximating general nonlinear time-dependent Schr\"odinger equations which may be high-dimensional. Our formulation integrates both the Fisk-Stratonovich and It\^o integrals within the framework of backward stochastic differential equations (BSDEs). Utilizing this Feynman-Kac representation, we propose learning-based approaches for numerical approximations. To demonstrate the accuracy and effectiveness of the proposed method, we conduct numerical experiments in both low- and high-dimensional settings, complemented by a convergence analysis. These results address the open problem concerning deep-BSDE methods for numerical approximations of high-dimensional time-dependent nonlinear Schr\"odinger equations (cf. [Proc. Natl. Acad. Sci. 15 (2018), pp. 8505-8510] and [Frontiers Sci. Awards Math. (2024), pp. 1-14] by Han, Jentzen, and E).

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arXiv2023

Viscosity Solutions of a class of Second Order Hamilton-Jacobi-Bellman Equations in the Wasserstein Space

This paper is devoted to solving a class of second order Hamilton-Jacobi-Bellman (HJB) equations in the Wasserstein space, associated with mean field control problems involving common noise. The well-posedness of viscosity solutions to the HJB equation under a new notion is established under general assumptions on the coefficients. Our approach adopts the smooth metric developed by Bayraktar, Ekren, and Zhang [Proc. Amer. Math. Soc. (2023)] as our gauge function for the purpose of smooth variational principle used in the proof of comparison theorem. Further estimates and regularity of the metric, including a novel second order derivative estimate with respect to the measure variable, are derived in order to ensure the uniqueness and existence.

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arXiv2023

A Viscosity Solution Theory of Stochastic Hamilton-Jacobi-Bellman equations in the Wasserstein Space

This paper is devoted to a viscosity solution theory of the stochastic Hamilton-Jacobi-Bellman equation in the Wasserstein spaces for the mean-field type control problem which allows for random coefficients and may thus be non-Markovian. The value function of the control problem is proven to be the unique viscosity solution. The major challenge lies in the mixture of the lack of local compactness of the Wasserstein spaces and the non-Markovian setting with random coefficients and various techniques are used, including Ito processes parameterized by random measures, the conditional law invariance of the value function, a novel tailor-made compact subset of measure-valued processes, finite dimensional approximations via stochastic n-player differential games with common noises, and so on.

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