连续时间马氏决策过程若干新准则的研究

Markov decision processes have been widely applied in many areas, such as finance, insurance, risk management, communication network, inventory management and queueing systems. The expected utility theory and cumulative prospect theory are applied to continuous-time Markov decision processes and a research will be carried out from the two viewpoints of the risk and the behaviors of the decision-makers. We will study the following problems in this project. . (1) The utility functions are used to describe the decision-makers’ risk attitude and we will consider the expected utility function criterion under the unbounded transition rates and the general convex or concave utility functions. . (2) We will make an attempt to study the dynamic risk measures of continuous-time Markov decision processes and their relative properties. Moreover, we will consider the mean-risk criterion under a class of risk measures with the property of law invariance. More precisely, the optimality criterion to be considered is to find a policy which maximizes the expected payoffs over the set of all feasible policies satisfying the constraint that the risk of the payoffs does not exceed a given constant. . (3) The probability distortion functions are used to describe the behavior that the decision-makers enlarge a small probability or diminish a large probability, and we will study the semi-variance minimization criterion under the probability distortion. . We will investigate the existence of optimal policies and give the effective methods of the approximate computations of the optimal policies for the above new criteria. The research of this project is of important significance on both theoretical and applied sides. It not only enriches the decision theory of the stochastic dynamic systems, but also provides a theoretical basis for the applied research, such as Markov decision models with applications to risk management.

马氏决策过程已在金融保险、风险管理、通信网络、库存管理、排队系统等众多领域有广泛的应用。本项目将期望效用函数理论和累积前景理论应用到连续时间马氏决策过程，从风险和决策者的行为这两个视角研究以下问题：(1) 利用效用函数刻画决策者的风险态度，考虑转移率无界且效用函数为一般的凸函数或凹函数的期望效用函数准则；(2) 探索连续时间马氏决策过程的动态风险度量及其相关性质，并对满足分布不变性的一类风险度量，考虑均值-风险准则，即在效益的风险不超过给定常数的策略类中寻找使得效益的均值最大的策略；(3) 通过概率扭曲函数刻画决策者人为地放大小概率事件或缩小大概率事件的行为，考虑加入概率扭曲函数的半方差准则。我们将研究上述新的准则最优策略的存在性及其有效的近似计算方法。本项目的研究在理论上和应用上都具有重要意义，不仅能丰富随机动态系统的决策理论，而且为马氏决策模型在风险管理等方面的应用研究提供理论依据。

本项目主要研究受控连续时间跳过程的期望效用函数准则和均值-半方差准则等若干新准则，具体如下：. （1）由正则效用函数诱导的平均费用准则（简称U-平均费用准则）最优策略的存在性及其计算方法。关于该准则，给出了最优策略的存在性条件，证明了最优值函数关于风险灵敏性系数的连续性，建立了U-平均费用准则与风险灵敏性平均费用准则、期望平均费用准则之间的联系，得到了U-平均最优平稳策略的存在性和计算最优策略的方法。.（2）由指数效用函数诱导的风险灵敏性平均费用准则最优策略的存在性。关于该准则，建立了风险灵敏性平均最优不等式解的存在性，证明了最优策略的存在性，给出了simultaneous Doeblin假设可验证的充分条件和风险灵敏性平均最优不等式严格成立的例子。.（3）均值-半方差准则最优策略的存在性及其计算方法。关于该准则，证明了折扣总费用半方差问题的最优值函数是均值-半方差准则最优方程的解，得到了最优策略的存在性，给出了近似计算最优值函数和最优策略的值迭代算法并分析了该算法的收敛性。.（4）风险灵敏性有限阶段准则零和博弈和非零和博弈问题均衡点的存在性、无约束期望平均准则和受约束期望平均准则非零和博弈问题纳什均衡点的存在性。关于这些博弈问题，给出了均衡点的存在性条件并证明了其存在性。. 本项目关于U-平均费用准则和均值-半方差准则等若干新准则所取得的理论结果为受控连续时间跳过程的应用研究奠定了理论基础。

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