带切换过程(系统)的稳定性及收敛速率估计

带切换过程(系统)是一类连续状态与离散状态过程的混合系统，可提供更贴合实际的数学模型，在金融、自动控制、生物学等领域有广泛的应用。稳定性分析是带切换过程模型研究中的一个基本课题，大量例子表明带切换过程稳定性对切换机制和噪音干扰这两个因素的影响很“敏感”，而其一般理论待进一步发展和验证。稳定性收敛速率的主阶界定和定量分析在算法设计等应用中有重要作用，而针对当前模型的这方面研究还很有限。.本项目拟研究(1)切换机制、噪音干扰对带切换过程稳定性的影响，包括(半)马氏切换机制、逗留时长服从某分布的切换机制等结合布朗运动、列维过程等噪音对常见稳定性的影响；(2)带切换过程稳定性的收敛速率，包括界定收敛速率主阶的定性研究及定量估计。.本项目研究难点是如何将随机微分方程和跳过程的研究工具有效结合，区分它们在混合系统演变过程中的不同作用，给出该模型新的有效的刻画系统发展的方法。

Regime-switching stochastic process is a kind of mixed systems combining continuous state processes with discrete state processes. It can provide more practical models in various applications, which is widely used in finance, automatic control and biology, etc.. The analysis of the stability is basic for the Regime-switching stochastic process. A large number of examples show that whether the Regime-switching stochastic process is stability or not is sensitive to the two factors including switching mechanism and noise interference. The general theory waits for further development and verification. The study of the stability of the convergence rate plays an important role in the design of Algorithms and other applications, but the current research on such aspect is very limited. .The program is proposed to investigate the contents as follows:(1) We will analysis the influence of switching mechanism and noise on the common stability of the Regime-switching stochastic process. Here the switching mechanisms include that semi-Markovian or Markovian cases or its sojourn time of the switching mechanism is subjected to some specific distribution etc., and the noise includes BM motion, Levy process and so on. (2) We will investigate the convergence rate of its stability, including qualitative analysis and quantitative estimation of the main order of the convergence rate. .The difficulty of the study is to consider how to effectively combine the research tools of diffusion processes and jump processes, distinguishing their different roles in the evolution of mixed systems, presenting new and effective methods for describing the development of the system.