Collaborative Research: Infinite horizon risk-sensitive control of diffusions with applications in stochastic networks

该研究将推动随机控制的数学分析，并对应用概率和随机网络做出重要贡献。该研究还将对大规模数据中心、制造业、电信、医疗保健、库存和服务系统中的实际应用产生影响，提供有效管理它们的技能和工具。这样的系统通常可以建模为随机网络，具有多个作业和许多服务器，以及复杂的网络拓扑结构。这些复杂的网络系统在各种随机环境下的运行和管理受到许多风险因素的影响。本研究将开发先进的方法和算法，以提供解决方案，减轻大规模网络模型系统中潜在的操作风险。该模型系统大致描述了大规模并行服务器网络中的系统动力学。该研究将提供近似最优调度和其他操作策略。风险敏感控制的优点是在存在干扰和不确定性的情况下仍能取得良好的性能。它还限制了大的波动，因为它惩罚了运行成本的较高时刻。研究人员将把他们的研究结果纳入现有的随机网络和控制研究生课程，并通过有关研究课题的研讨会加以传播。这项研究涉及一组跨学科的研究人员，其中包括那些来自代表性不足的少数群体的研究人员，并为具有新数学技能的研究生提供培训机会。本文的研究目标是：(1)建立一个研究椭圆系统特征值和积分微分算子的综合理论框架，以解决状态切换和跳跃扩散的无限视界风险敏感控制（IHRS）中的相关问题。(2)发展建立渐近最优性所需的技术，并研究相关的随机微分对策和大偏差表征。(3)研究了大时间渐近行为和相对值迭代算法，这是滚动地平线控制和强化学习方法的基础。这项研究将极大地推进积分微分算子和椭圆系统的特征值理论，并为扩散（带跳跃）和状态切换扩散的风险敏感控制产生开创性的方法。在分析方面，本研究将极大地促进目前文献中关于无界域中非线性特征值问题的努力。期望在非对称算子的变分特征、最大和大偏差原理以及相关的Feynman-Kac半群方面得到丰富的结果。该研究的另一个重要贡献是分析了大时间渐近行为，其中包括相对值迭代算法和滚动地平线控制的研究。该研究还将促进对大规模并行服务器网络的风险敏感渐近最优调度策略的理解，包括那些在随机环境中引起跳跃扩散和状态切换扩散限制的调度策略。涉及等效随机微分对策和空间截断技术的新方法将被开发来证明渐近最优性的下界和上界。最后，但并非最不重要的是，本研究旨在缩小概率和分析方法之间的差距，旨在改善两个社区之间的互动。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This research will advance mathematical analysis in stochastic control and make important contributions to applied probability and stochastic networks. The research will also have an impact on real-world applications in large-scale data centers, manufacturing, telecommunications, healthcare, inventory, and service systems, providing skills and tools to manage them effectively. Such systems can often be modeled as a stochastic network, with multiple jobs and many servers, and complex network topology. The operations and management of these sophisticated networked systems are subject to many risk factors under various random environments. This research will develop advanced methods and algorithms to provide solutions that mitigate the potential operational risks in a large-scale network model system. The model system roughly describes the system dynamics in large-scale parallel server networks. The research will provide approximate optimal scheduling and other operational policies. Risk-sensitive control has the advantage of achieving good performance in the presence of disturbances and uncertainty. It also limits large fluctuations since it penalizes higher moments of the running cost. The investigators will incorporate their findings into the existing graduate courses in stochastic networks and control, and disseminate them through seminars on relevant research topics. The research involves a team of interdisciplinary researchers, including those from underrepresented minority groups, and provides training opportunities for graduate students with new mathematical skills. The objectives of the research are: (1) To develop a comprehensive theoretical framework for the study of eigenvalues of elliptic systems and integro-differential operators to address the associated problems in infinite-horizon risk sensitive control (IHRS) of regime-switching and jump diffusions. (2) To develop the techniques required to establish asymptotic optimality and study the associated stochastic differential games and large deviation characterizations. (3) To study the large-time asymptotic behavior and relative value iteration algorithms, which form the basis of rolling horizon control and reinforcement learning methods. This research will greatly advance the theory of eigenvalues of integro-differential operators and elliptic systems and produce ground-breaking methodologies for risk-sensitive control of diffusions (with jumps) and regime-switching diffusions. On the analytical side, this research will greatly contribute to the current efforts in the literature concerning nonlinear eigenvalue problems in unbounded domains. A wealth of results on variational characterizations, maximum and large deviation principles, and the associated Feynman-Kac semigroup for nonsymmetric operators are expected to be obtained. Another important contribution of the proposed research is analyzing large-time asymptotic behavior, which includes the study of relative value iteration algorithms and rolling horizon control. The research will also advance the understanding of the risk-sensitive asymptotically optimal scheduling policies for large-scale parallel server networks, including those in random environments that give rise to jump-diffusion and regime-switching diffusion limits. New methods involving the equivalent stochastic differential game and spatial truncation techniques will be developed to prove lower and upper bounds for asymptotic optimality. Last, but not least, this research aims to close the gap between probabilistic and analytical methods, aiming to improve the interaction between the two communities.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.