Given a closed, bounded convex set $\mathcal{W}\subset{\mathbb {R}}^d$ with nonempty interior, we consider a control problem in which the state process $W$ and the control process $U$ satisfy \[W_t= w_0+\int_0^t\vartheta(W_s) ds+\int_0^tσ(W_s) dZ_s+GU_t\in \mathcal{W},\qquad t\ge0,\] where $Z$ is a standard, multi-dimensional Brownian motion, $\vartheta,σ\in C^{0,1}(\mathcal{W})$, $G$ is a fixed matrix, and $w_0\in\mathcal{W}$. The process $U$ is locally of bounded variation and has increments in a given closed convex cone $\mathcal{U}\subset{\mathbb{R}}^p$. Given $g\in C(\mathcal{W})$, $κ\in{\mathbb{R}}^p$, and $α>0$, consider the objective that is to minimize the cost \[J(w_0,U)\doteq\mathbb{E}\biggl[\int_0^{\infty}e^{-αs}g(W_s) ds+\int_{[0,\infty)}e^{-αs} d(κ\cdot U_s)\biggr]\] over the admissible controls $U$. Both $g$ and $κ\cdot u$ ($u\in\mathcal{U}$) may take positive and negative values. This paper studies the corresponding dynamic programming equation (DPE), a second-order degenerate elliptic partial differential equation of HJB-type with a state constraint boundary condition. Under the controllability condition $G\mathcal{U}={\mathbb{R}}^d$ and the finiteness of $\mathcal{H}(q)=\sup_{u\in\mathcal{U}_1}\{-Gu\cdot q-κ\cdot u\}$, $q\in {\mathbb{R}}^d$, where $\mathcal{U}_1=\{u\in\mathcal{U}:|Gu|=1\}$, we show that the cost, that involves an improper integral, is well defined. We establish the following: (i) the value function for the control problem satisfies the DPE (in the viscosity sense), and (ii) the condition $\inf_{q\in{\mathbb{R}}^d}\mathcal{H}(q)<0$ is necessary and sufficient for uniqueness of solutions to the DPE. The existence and uniqueness of solutions are shown to be connected to an intuitive ``no arbitrage'' condition. Our results apply to Brownian control problems that represent formal diffusion approximations to control problems associated with stochastic processing networks.
Extended mean-field control problems with Poissonian common noise: Stochastic maximum principle and Hamiltonian-Jacobi-Bellman equation
Xidian University（西安电子科技大学）；University of Science and Technology of China（中国科学技术大学）；Harbin Institute of Technology（哈尔滨工业大学）；The Hong Kong Polytechnic University（香港理工大学）
Controlled Interacting Branching Diffusion Processes: A Viscosity Approach
Centre de Recherche en Économie et de Statistiques (CREST), Groupe ENSAE-ENSAI, ENSAE Paris, Institut Polytechnique de Paris（经济与统计研究中心（CREST），ENSAE-ENSAI小组，巴黎ENSAE，巴黎理工学院）
Ergodic control of diffusions with compound Poisson jumps under a general structural hypothesis
研究了一类由复合泊松过程驱动的受控跳跃扩散过程的遍历控制问题。这扩展了[SIAM J. Control Optim. 57（2019），No. 2，1516-1540]到不是接近单调的运行成本。这种普遍性是需要在应用程序，如大规模并行服务器网络的优化调度。我们通过Hamilton-Jacobi-Bellman（HJB）方程提供了一个完整的最优性表征，在温和的假设下，我们还表现出解的正则性。此外，我们还证明了最优平稳马尔可夫控制是a.s.路径最优最后，我们证明了在一个足够大的有界域上求解HJB，可以在紧集之外固定一个稳定控制，并获得近最优解。这对于构造多类并行服务器网络的渐近最优调度策略是有用的。
