本项目拟研究几类随机观察风险模型的税收与最优分红问题，这是风险理论中的最新热点课题，也是随机过程理论与金融保险领域的交叉研究。“随机观察”是近几年新引入的概念，指的是保险公司定期检查公司的财务状况，而不是连续地监测。在随机观察框架下建立的风险模型，具有很好的实际意义与研究价值，以此为出发点，本项目将研究以下问题：一、将loss-carry-forward税收因素融入到随机观察风险模型的构建中，考虑破产概率、Gerber-Shiu函数、期望折现税收和最优收税边界；二、研究随机观察风险模型中的最优分红问题，利用HJB方程的方法得到“随机分红”问题的最优分红策略，还涉及破产概率、期望罚金函数、期望折现分红及最优分红边界；三、结合过程本身的性质研究逐段决定马尔可夫过程的实质性破产问题，讨论实质性破产概率和负持续时的相关性质。本项目的研究符合当前保险事业的发展需求，将促进随机过程理论的实际应用。

This project will study taxation and optimal dividend problems for some kinds of risk models with randomized observation periods. It is the latest topic in the risk theory, but also cross-over study in the risk theory and financial insurance field. The concept of randomized observation periods has been introduced in recent years, meant that the insurance company checks the balance on a periodic basis rather than continuous observations. The risk model in the framework of stochastic observations has practical and theoretical value. To this end, this project will study the following problems: first, we embed loss-carry-forward taxation and random observations into the risk model, and consider the ruin probability, the Gerber-Shiu function, the expected discounted tax payments and the optimal taxation strategy. Second, we study the optimal dividend problem of risk models with random observations, found the optimal dividend strategy for the random dividend problem by HJB equation, and obtain the ruin probability, the expected discounted penalty function, the expectation of discounted dividends and the optimal dividend barrier. Finally, we investigate the bankruptcy problem in the piecewise-deterministic Markov process by their properties, and discuss relative properties of the bankruptcy property and the total duration of the negative surplus. The research of this project conforms to the development of insurance undertakings, and will promote the practical application of stochastic process theory.