半马氏随机对策的折扣概率准则及其应用研究

Semi-Markov stochastic games have been widely stuided, where the current literature focuses on expected criteria in theory and there is a few applications on semi-Markov stochatic games. As we know, the optimality under expected criteria is to optimize the expectation of the reward or cost. Based on the existing work, this proposal includes two parts, i.e., theory and applications. For theory, under discounted probability criteria, we make all players earn maximum security by designing policy vectors such as marketing plans in business, i.e., we attempt to give the conditions for the existence of optimal policy vectors and the algorithm. The motivation of this part stems from the following actual cases: a) the risk-sensitive are more concerned with their risk probability than the expectation of rewards; b) in business systems, players aim to optimize their annual reward or cost incurred before the first profit, and thus the horizon of decision can be finite or randomized; c) the discount factor，e.g., corresponding to an interest rate which varies with states in financial systems, should be state-dependent . For applications, after estabilishing semi-Markov stochastic game models for business and finance systems, we try to use part one's results to choose marketing plans in business and portfolio strategies in finance and so on, which leads to the minimum risk for all players. The contents above are new, which is to promote new developments in both theoretical and applied researches of semi-Markov stochastic games, and is in favor of the usage of theory in real-world situations.

现有文献对半马氏随机对策的理论研究局限于探讨期望准则，且应用研究较少，目标是优化局中人的期望收益或成本。本项目拟在已有工作的基础上，开展半马氏随机对策的理论和应用两大研究。理论研究为在折扣概率准则下设计策略向量（如商业系统的营销方案）使局中人的安全均达到最优，即寻求最优策略向量存在的条件及算法，其中决策区间有限或随机以及折扣因子依赖状态，该部分的提出源于下列事实：a)风险敏感者更加关注收益的风险程度而不是期望收益；b)决策区间可能是有限或随机的（如商业系统年度考核、首次盈利）；c)折扣因子可能是依赖状态的（如金融系统的利率）。应用研究是为商业系统和金融系统等建立半马氏随机对策模型，并运用上述理论成果指导商业系统的营销方案和金融系统的投资方式等的选取，使得局中人承担最小的风险。该研究内容在半马氏随机对策中是新的，不仅能推动半马氏随机对策理论的新进展，还能扩大其应用范围，有利于理论指导实践。

在当前国际国内互利共赢的背景下，本项目以非零和随机对策模型为基础，对概率准则和期望折扣准则下的相关问题进行了深入地研究，具体如下：(1) 对概率准则的探讨：在离散时间两人随机对策模型框架下，通过分析概率准则的一些重要性质，得到了最优值函数的迭代计算方法，并在较弱的条件下，证明了相应的最优方程和纳什均衡的存在性。最后，针对排队系统实例，我们建立了离散时间两人非零和随机对策模型，并在基于模型原始数据的假设条件下证明了纳什均衡的存在性。相应结果发表在期刊《Dynamic Games and Applications》上。(2) 对期望折扣准则的研究：在连续时间N人随机对策模型框架下，通过寻找期望折扣准则有限的条件及分析它的特征，其中折扣因子依赖状态，建立了Shapley方程，在更广泛地随机历史依赖策略类内得到了纳什均衡存在的条件，并说明了两人零和随机对策相应结论是我们模型特殊情况下的直接结果。最后，我们对金融系统实例建立了连续时间N人非零和随机对策模型，给出了纳什均衡存在的条件。相关工作已经发表在期刊《IEEE Transactions on Automatic Control》上。. 上述(1)中概率准则在随机对策模型中的讨论是马氏决策过程下相应模型的延伸和拓展，是评估风险的准则在随机对策中的发展。上述(2)首次探讨具有依赖状态折扣因子和随机历史依赖策略的随机对策模型，它们能够分别较好地描述实际经济活动中利率的不确定性和财富状况的不平衡性以及历史信息对决策者行动选取的影响。故本课题研究成果具有重要的理论与应用价值。