Nonlinear dynamic optimization theory on stochastic model and its application to mathematical finance

研究成果总结如下：1.研究成果综述。我们考虑多阶段决策过程，其中准则函数是极小值函数的期望，并将其表示为具有嵌入参数的马尔可夫决策过程。该策略依赖于包括过去嵌入参数的历史，并且每个阶段的奖励是随机的，并且取决于当前状态、当前动作和下一状态。利用算子给出了最优性方程，证明了存在依赖于当前状态和嵌入参数的右连续确定性马尔可夫策略。我们考虑具有目标集的马尔可夫决策过程，其中准则函数是极小函数的期望。我们将问题表示为具有递归类的无限水平情形。证明了在一定条件下，最优值函数是最优性方程的唯一解，并且存在平稳最优策略。并给出了政策改进的方法。我们认为…更确切地说，这是一个具有结合准则的随机最短路径问题，其中对于图的每个节点，我们选择后继节点集合上的概率分布，以便最优地到达给定的目标节点。我们将这类问题描述为一个结合马尔可夫决策过程。证明了最优值函数是最优性方程的唯一解，并找到了最优平稳策略。给出了一种数值迭代方法和一种策略改进方法。我们考虑效用约束的马尔可夫决策过程。在多个期望效用约束下，总贴现报酬的期望效用最大化。通过引入相应的拉格朗日函数，导出了效用约束最优化的鞍点定理。约束最优策略的存在是由参数效用指定的最优行动集来刻画的。我们考虑一个不等条件，其中一方大于或等于另一方的倍数，且等式成立的充要条件是一个值是另一个变量的倍数。我们证明了一元函数的四对Golden不等式之间的交叉对偶。较少

The summary of research results is as follows.1. We consider multistage decision processes where a criterion function is an expectation of minimum function and formulate it as Markov decision processes with imbedded parameters. The policy depends upon a history including past imbedded parameters and the rewards at each stage is random and depends upon a current state, a current action and a next state. We give an optimality equation by using operators and show that there exist a right continuous deterministic Markov policy which depend upon a current state and an imbedded parameter.2. We consider Markov decisions processes with a target set, where criterion function is an expectation of minimum function. We formulate the problem as an infinite horizon case with a recurrent class. We show under some conditions that an optimal value function is a unique solution to an optimality equation and there exists an stationary optimal policy. Also we give a policy improvement method.3. We conside … More r a stochastic shortest path problem with associative criteria in which for each node of a graph we choose a probability distribution over the set of successor nodes so as to reach a given target node optimally. We formulate such a problem as an associative Markov decision processes. We show that an optimal value function is a unique solution to an optimality equation and find an optimal stationary policy. Also we give a value iteration method and a policy improvement method.4. We consider utility-constrained Markov decision processes. The expected utility of the total discounted reward is maximized subject to multiple expected utility constraints. By introducing a corresponding Lagrange function, saddle-point theorem of the utility constrained optimization is derived. The existence of a constrained optimal policy is characterized by optimal action sets specified with a parametric utility.5. We consider an inequality condition where one side is greater than or equal to a multiple of the other side and an equality holds if and only if one value is a multiple of the other variable. We show a cross-duality between four pairs of Golden inequalities for one-variable functions. Less