Analysis, Simulation, and Applications of Stochastic Systems

随机系统是随机干扰起重要作用的系统。源于新兴和现有的应用在网络系统，无线通信，信号处理，经济学和生态学，该项目包括动态演变的随机系统与不确定性，不同的配置之间的切换，和复杂的结构的研究。感兴趣的网络系统包括金融，通信，社会，生物和生态网络。这项工作将致力于学习随机网络系统的内在属性，开发数学模型和新的数学方法来分析这样的系统，并设计有效的计算方案，优化和控制这样的系统，以满足预期的目标。研究结果对经济学、非线性系统辨识与估计、无人驾驶汽车及其他多智能体系统、生态系统中的生物多样性以及社交网络等领域的研究具有重要的应用价值。该项目将涉及本科生和研究生，并将研究与教学和学生培训相结合。这项工作将有助于进一步发展数学理论，计算方法和应用，并改善数学教育。本研究课题的目的是为了满足各种各样的应用需求，主要研究以下课题。（1）将开发和分析具有随机切换的随机模型。系统的新特征包括：（i）具有可数状态空间的依赖于过去的切换，以及（ii）具有非局部算子、有限切换集和sigma有限跳跃测度的切换跳跃扩散。将获得复发、阳性复发和遍历性的标准。(2)我们将研究白色杂讯扰动下的Kolmogorov型系统，其中扩散是退化的。将研究控制依赖环境保护区、传染病和生态的应用。(3)新的算法切换扩散和随机逼近将开发和他们的收敛速度进行研究。(i)对于Milstein型算法的开关扩散的解决方案，它将被证明，该算法保持1阶收敛速度作为他们的扩散对应。(ii)受多智能体系统、共识和群集应用的启发，随机近似算法的新颖性包括包含状态相关切换、状态相关观测噪声和一般时间相关非线性函数。(4)本文给出了Hammerstein非线性系统的精确误差估计。它将被证明，估计逃脱一个小的邻域的真参数的概率是指数小。(5)在强逼近意义下，得到了重复删除随机网络逼近格式的精确误差界。这将对随机动态图的研究及其在社交网络中的应用产生影响。将进行大量的数值实验和模拟，以补充分析和算法设计。这些项目将涉及本科生和研究生的参与。

Stochastic systems are systems in which random disturbances play a significant role. Stemming from emerging and existing applications in networked systems, wireless communications, signal processing, economics, and ecology, this project encompasses the study of dynamically evolving stochastic systems with uncertainties, switching among different configurations, and complex structures. The networked systems of interest include financial, communication, social, biological, and ecological networks. The work will be devoted to learning the intrinsic properties of stochastic network systems, developing mathematical models and novel mathematical methods for analyzing such systems, and designing efficient computational schemes for optimization and control of such systems to meet desired goals. The results of the research will be useful for applications to economics, nonlinear system identification and estimation, un-manned vehicles and other multi-agent systems, biodiversity in ecological systems, and social networks. This projects will involve undergraduate and graduate students and will integrate the research with teaching and student training. This work will contribute jointly to the further development of mathematical theory, computational methods and applications, and the improvement of mathematics education. Motivated by a wide variety of applications, this project will study the following research topics. (1)Stochastic models with random switching will be developed and analyzed. Novel features of the systems include (i) past-dependent switching having a countable state space, and (ii) switching jump diffusions with non-local operators, finite switching set, and sigma finite jump measures. Criteria for recurrence, positive recurrence, and ergodicity will be obtained. (2) Kolmogorov-type systems under white noise perturbations, where the diffusions are degenerate, will be investigated. Applications to control dependent environmental protection zones, infectious disease and ecology will be studied. (3) New algorithms for switching diffusions and stochastic approximation will be developed and their rates of convergence will be studied. (i) For Milstein-type algorithms for solutions of switching diffusions, it will be shown that the algorithms preserve order 1 convergence rates as their diffusion counterpart. (ii) Motivated by applications to multi-agent systems, consensus, and swarming, the novelties of the stochastic approximation algorithms include the inclusion of state-dependent switching, state-dependent observation noise, and general time-dependent nonlinear functions. (4) Precise error estimates for identification of Hammerstein nonlinear systems with quantized observations will be obtained. It will be proved that the estimates escape from a small neighborhood of the true parameter with a probability that is exponentially small. (5) Accurate error bounds for approximation schemes of duplication-deletion random networks in the sense of strong approximation will be obtained. This will have impact on the study of random dynamic graphs and applications to social networks. Extensive numerical experiments and simulations will be performed to complement the analysis and algorithm design. The projects will involve the participation of undergraduate and graduate students.