Kengo Terai
This paper is concerned with the asymptotic analysis of infinite systems of weakly coupled stationary Hamilton-Jacobi-Bellman equations as the discount factor tends to zero. With a specific Hamiltonian, we show the convergence of the solution and prove the solvability of the corresponding ergodic problem.
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Solution Theory of Hamilton-Jacobi-Bellman Equations in Spectral Barron Spaces
本文研究全空间静态HJB方程在谱Barron空间中的解理论，证明在系数为谱Barron函数且折扣因子足够大时解可被一致有界序列局部一致逼近，从而可用两层神经网络无维数灾难地近似求解。
For a class of Bellman equations in bounded domains we prove that sub- and supersolutions whose growth at the boundary is suitably controlled must be constant. The ellipticity of the operator is assumed to degenerate at the boundary and a condition involving also the drift is further imposed. We apply this result to stochastic control problems, in particular to an exit problem and to the small discount limit related with ergodic control with state constraints. In this context, our condition on the behavior of the operator near the boundary ensures some invariance property of the domain for the…
Projected Evolutionary Lifting and Well-Posedness of Stationary Hamilton-Jacobi-Bellman Equations in Infinite Dimensions
本文研究了无限维 stationary Hamilton-Jacobi-Bellman 方程的适定性，通过投影演化提升技术解决了无全局平滑性质、奇异动力学和状态依赖运行成本等挑战，证明了任意折扣率下的解存在性与唯一性。
We consider stochastic differential games with $N$ nearly identical players, linear-Gaussian dynamics, and infinite horizon discounted quadratic cost. Admissible controls are feedbacks for which the system is ergodic. We first study the existence of affine Nash equilibria by means of an associated system of $N$ Hamilton-Jacobi-Bellman and $N$ Kolmogorov-Fokker-Planck partial differential equations, proving that for small discount factors quadratic-Gaussian solutions exist and are unique. Then, we prove the convergence of such solutions to the unique quadratic-Gaussian solution of the pair of M…
Remarks on selection problems for first-order discounted mean field games
研究一阶稳态平均场博弈中消失折扣极限的选择问题，通过状态空间分解证明选择值函数在间隙残差集上可能非唯一，并给出唯一性和收敛性条件。
Global Sobolev convergence of Howard iteration for parabolic Bellman equations with controlled diffusion
本文证明带受控扩散的抛物型Bellman方程在一般条件下Howard策略迭代的全局Sobolev收敛，并推广至多维及熵正则化模型，关键是通过残差测度收敛构造唯一强解。
This is the first of a series of papers on the interior regularity of fully nonlinear degenerate elliptic equations. We consider a stochastic optimal control problem in which the diffusion coefficients, drift coefficients and discount factor are independent of the spacial variables. Under suitable assumptions, for $k=0,1$, when the terminal and running payoffs are globally $C^{k,1}$, we obtain the $C^{k,1}$-smoothness of the value function, which yields the existence and uniqueness of the solution to the associated Dirichlet problem for the degenerate Bellman equation.
We study the well-posedness of Hamilton-Jacobi-Bellman equations on subsets of $\mathbb{R}^d$ in a context without boundary conditions. The Hamiltonian is given as the supremum over two parts: an internal Hamiltonian depending on an external control variable and a cost functional penalizing the control. The key feature in this paper is that the control function can be unbounded and discontinuous. This way we can treat functionals that appear e.g. in the Donsker-Varadhan theory of large deviations for occupation time measures. To allow for this flexibility, we assume that the internal Hamiltoni…
Policy Iteration for Exploratory Hamilton--Jacobi--Bellman Equations
本文研究大折扣率下熵正则化随机控制问题的策略迭代算法，分别在有界系数（扩散项控制满足小性条件）和无界系数（无扩散控制）两种场景下，通过建立值序列的$\mathcal{C}^{2,α}$和$\mathcal{C}^{1,α}$估计证明其收敛性。
C^(sigma+alpha) regularity for fractional Hamilton-Jacobi equations, and rates of convergence in periodic homogenization
该研究针对非局部贝尔曼型方程，建立其C^(σ+α)正则性理论，结合折扣近似校正子量化扰动检验函数方法，证明分数阶哈密顿-雅可比方程周期均匀化解以ε^(σα/(2+α))的速率收敛到有效解。
Path-dependent Hamilton-Jacobi equations with u-dependence and time-measurable Hamiltonians
本文建立了一类一般路径依赖Hamilton-Jacobi方程极小极大解的存在唯一性，适用于含折扣因子和时间可测数据的延迟泛函微分方程最优控制，并支撑了相关随机控制研究。
Continuous time Markov chain based approximation of stationary and weak KAM Hamilton-Jacobi equations
本文用连续时间马尔可夫决策问题逼近有限维环面上的平稳与弱KAM Hamilton-Jacobi方程，通过Bellman方程和弱KAM理论获得有效Hamiltonian的逼近，并证明收敛性，最终得到代数方程组形式的数值格式。
Equilibrium under Time-Inconsistency: A New Existence Theory by Vanishing Entropy Regularization
本文通过消失熵正则化方法，解决了时间不一致随机控制问题中均衡哈密顿-雅可比-贝尔曼方程解的存在性问题，无需依赖传统严格正则性假设。
The purpose of this work is to introduce a notion of weak solution to the master equation of a potential mean field game and to prove that existence and uniqueness hold under quite general assumptions. Remarkably, this is achieved without any monotonicity constraint on the coefficients. The key point is to interpret the master equation in a conservative sense and then to adapt to the infinite dimensional setting earlier arguments for hyperbolic systems deriving from a Hamilton-Jacobi-Bellman equation. Here, the master equation is indeed regarded as an infinite dimensional system set on the spa…
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