Searcharxiv⌕ Search

arXiv subjects

Chenchen Mou

Publications and source records attributed to Chenchen Mou.

32 records · Page 2Linked to original sources
arXiv2022

Mean Field Games of Controls: Propagation of Monotonicities

The theory of Mean Field Game of Controls considers a class of mean field games where the interaction is through the joint distribution of the state and control. It is well known that, for standard mean field games, certain monotonicity condition is crucial to guarantee the uniqueness of mean field equilibria and then the global wellposedness for master equations. In the literature, the monotonicity condition could be the Lasry-Lions monotonicity, the displacement monotonicity, or the anti-monotonicity conditions. In this paper, we investigate all these three types of monotonicity conditions for Mean Field Games of Controls and show their propagation along the solutions to the master equations with common noises. In particular, we extend the displacement monotonicity to semi-monotonicity, whose propagation result is new even for standard mean field games. This is the first step towards the global wellposedness theory for master equations of Mean Field Games of Controls.

math.PR↗
arXiv2022

Linear-Quadratic Mean Field Games of Controls with Non-Monotone Data

In this paper, we study a class of linear-quadratic (LQ) mean field games of controls with common noises and their corresponding $N$-player games. The theory of mean field game of controls considers a class of mean field games where the interaction is via the joint law of both the state and control. By the stochastic maximum principle, we first analyze the limiting behavior of the representative player and obtain his/her optimal control in a feedback form with the given distributional flow of the population and its control. The mean field equilibrium is determined by the Nash certainty equivalence (NCE) system. Thanks to the common noise, we do not require any monotonicity conditions for the solvability of the NCE system. We also study the master equation arising from LQ mean field games of controls, which is a finite-dimensional second-order parabolic equation. It can be shown that the master equation admits a unique classical solution over an arbitrary time horizon without any monotonicity conditions. Beyond that, we can solve the $N$-player games directly by further assuming the non-degeneracy of the idiosyncratic noises. As byproducts, we prove the quantitative convergence results from the $N$-player game to the mean field game and the propagation of chaos property for the related optimal trajectories.

math.OC↗
arXiv2022

Numerical methods for Mean field Games based on Gaussian Processes and Fourier Features

In this article, we propose two numerical methods, the Gaussian Process (GP) method and the Fourier Features (FF) algorithm, to solve mean field games (MFGs). The GP algorithm approximates the solution of a MFG with maximum a posteriori probability estimators of GPs conditioned on the partial differential equation (PDE) system of the MFG at a finite number of sample points. The main bottleneck of the GP method is to compute the inverse of a square gram matrix, whose size is proportional to the number of sample points. To improve the performance, we introduce the FF method, whose insight comes from the recent trend of approximating positive definite kernels with random Fourier features. The FF algorithm seeks approximated solutions in the space generated by sampled Fourier features. In the FF method, the size of the matrix to be inverted depends only on the number of Fourier features selected, which is much less than the size of sample points. Hence, the FF method reduces the precomputation time, saves the memory, and achieves comparable accuracy to the GP method. We give the existence and the convergence proofs for both algorithms. The convergence argument of the GP method does not depend on any monotonicity condition, which suggests the potential applications of the GP method to solve MFGs with non-monotone couplings in future work. We show the efficacy of our algorithms through experiments on a stationary MFG with a non-local coupling and on a time-dependent planning problem. We believe that the FF method can also serve as an alternative algorithm to solve general PDEs.

math.NA↗
arXiv2022

Mean Field Games Master Equations with Non-separable Hamiltonians and Displacement Monotonicity

In this manuscript, we propose a structural condition on non-separable Hamiltonians, which we term displacement monotonicity condition, to study second order mean field games master equations. A rate of dissipation of a bilinear form is brought to bear a global (in time) well-posedness theory, based on a--priori uniform Lipschitz estimates on the solution in the measure variable. Displacement monotonicity being sometimes in dichotomy with the widely used Lasry-Lions monotonicity condition, the novelties of this work persist even when restricted to separable Hamiltonians.

math.AP↗
arXiv2022

Wellposedness of Second Order Master Equations for Mean Field Games with Nonsmooth Data

In this paper we study second order master equations arising from mean field games with common noise over arbitrary time duration. A classical solution typically requires the monotonicity condition (or small time duration) and sufficiently smooth data. While keeping the monotonicity condition, our goal is to relax the regularity of the data, which is an open problem in the literature. In particular, we do not require any differentiability in terms of the measures, which prevents us from obtaining classical solutions. We shall propose three weaker notions of solutions, named as {\it good solutions}, {\it weak solutions}, and {\it weak-viscosity solutions}, respectively, and establish the wellposedness of the master equation under all three notions. We emphasize that, due to the game nature, one cannot expect comparison principle even for classical solutions. The key for the global (in time) wellposedness is the uniform a priori estimate for the Lipschitz continuity of the solution in the measures. The monotonicity condition is crucial for this uniform estimate and thus is crucial for the existence of the global solution, but is not needed for the uniqueness in such Lipschitz class. To facilitate our analysis, we construct a smooth mollifier for functions on Wasserstein space, which is new in the literature and is interesting in its own right. Following the same approach of our wellposedness results, we prove the convergence of the Nash system, a high dimensional system of PDEs arising from the corresponding $N$-player game, under mild regularity requirements. We shall also prove a propagation of chaos property for the associated optimal trajectories.

math.AP↗
arXiv2021

Robust exploratory mean-variance problem with drift uncertainty

We solve a min-max problem in a robust exploratory mean-variance problem with drift uncertainty in this paper. It is verified that robust investors choose the Sharpe ratio with minimal $L^2$ norm in an admissible set. A reinforcement learning framework in the mean-variance problem provides an exploration-exploitation trade-off mechanism; if we additionally consider model uncertainty, the robust strategy essentially weights more on exploitation rather than exploration and thus reflects a more conservative optimization scheme. Finally, we use financial data to backtest the performance of the robust exploratory investment and find that the robust strategy can outperform the purely exploratory strategy and resist the downside risk in a bear market.

math.OC↗
arXiv2019

Coupling Levy measures and comparison principles for viscosity solutions

We prove new comparison principles for viscosity solutions of non-linear integro-differential equations. The operators to which the method applies include but are not limited to those of Lévy-Itô type. The main idea is to use an optimal transport map to couple two different Lévy measures, and use the resulting coupling in a doubling of variables argument

math.AP↗
arXiv2018

Regularity theory for Second order Integro-PDEs

This paper is concerned with higher Hölder regularity for viscosity solutions to non-translation invariant second order integro-PDEs, compared to \cite{mou2018}. We first obtain $C^{1,α}$ regularity estimates for fully nonlinear integro-PDEs. We then prove the Schauder estimates for solutions if the equation is convex.

math.AP↗
arXiv2018

Existence of $C^α$ solutions to integro-PDEs

This paper is concerned with existence of a $C^α$ viscosity solution of a second order non-translation invariant integro-PDE. We first obtain a weak Harnack inequality for such integro-PDE. We then use the weak Harnack inequality to prove Hölder regularity and existence of solutions of the integro-PDEs.

math.AP↗
arXiv2018

Stochastic Representations for Solutions to Parabolic Dirichlet Problems for Nonlocal Bellman Equations

We prove a stochastic representation formula for the viscosity solution of Dirichlet terminal-boundary value problem for a degenerate Hamilton-Jacobi-Bellman integro-partial differential equation in a bounded domain. We show that the unique viscosity solution is the value function of the associated stochastic optimal control problem. We also obtain the dynamic programming principle for the associated stochastic optimal control problem in a bounded domain.

math.PR↗
arXiv2017

Geodesic of minimal length in the set of probability measures on graphs

We endow the set of probability measures on a weighted graph with a Monge--Kantorovich metric, induced by a function defined on the set of vertices. The graph is assumed to have $n$ vertices and so, the boundary of the probability simplex is an affine $(n-2)$--chain. Characterizing the geodesics of minimal length which may intersect the boundary, is a challenge we overcome even when the endpoints of the geodesics don't share the same connected components. It is our hope that this work would be a preamble to the theory of Mean Field Games on graphs.

math.CA↗
arXiv2017

Semiconcavity of viscosity solutions for a class of degenerate elliptic integro-differential equations in $\mathbb R^n$

This paper is concerned with semiconcavity of viscosity solutions for a class of degenerate elliptic integro-differential equations in $\mathbb R^n$. This class of equations includes Bellman equations containing operators of Lévy-Itô type. Hölder and Lipschitz continuity of viscosity solutions for a more general class of degenerate elliptic integro-differential equations are also provided.

math.AP↗
arXiv2017

Perron's method for nonlocal fully nonlinear equations

This paper is concerned with existence of viscosity solutions of non-translation invariant nonlocal fully nonlinear equations. We construct a discontinuous viscosity solution of such nonlocal equation by Perron's method. If the equation is uniformly elliptic in the sense of \cite{SS}, we prove the discontinuous viscosity solution is Hölder continuous and thus it is a viscosity solution.

math.AP↗