Analysis, Simulation, and Applications of Stochastic Systems

随机系统是随机扰动起重要作用的系统。这个项目源于网络系统、无线通信、信号处理、经济学和生态学中的新兴和现有应用，涵盖了具有不确定性、在不同配置之间切换和复杂结构的动态演化随机系统的研究。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性质，发展分析这类系统的数学模型和新的数学方法，并设计有效的计算方案来优化和控制这类系统，以达到预期的目标。研究结果将对经济学、非线性系统辨识和估计、无人飞行器和其他多智能体系统、生态系统中的生物多样性和社会网络的应用具有一定的参考价值。该项目将涉及本科生和研究生，并将把研究与教学和学生培训结合起来。这项工作将共同为数学理论、计算方法和应用的进一步发展以及数学教育的改进做出贡献。在广泛应用的推动下，本项目将研究以下研究课题。(1)建立和分析具有随机切换的随机模型。该系统的新特征包括(I)具有可数状态空间的过去依赖的切换，以及(Ii)使用非局部算子、有限切换集和sigma有限跳跃度量来切换跳跃扩散。将获得重现、正重现和遍历性的准则。(2)研究具有退化扩散的Kolmogorov型白噪声扰动系统。将研究在控制附属环境保护区、传染病和生态方面的应用。(3)将开发新的开关扩散算法和随机逼近算法，并研究它们的收敛速度。(I)对于开关扩散解的Milstein型算法，我们将证明该算法保持一阶收敛速度作为其相应的扩散项。(Ii)受多智能体系统、一致性和群体问题应用的启发，随机逼近算法的新颖性包括状态依赖的切换、状态依赖的观测噪声和一般的时间依赖的非线性函数。(4)给出了带量化观测值的Hammerstein非线性系统辨识的精确误差估计。将证明这些估计以指数小的概率逃离真实参数的一个小邻域。(5)得到了强逼近意义下去重随机网络逼近方案的精确误差界。这将对随机动态图的研究及其在社会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.