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

研究结果总结如下：1.我们考虑多阶段决策过程的准则函数是一个期望的最小函数，并制定它作为马尔可夫决策过程的嵌入参数。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，并证明了存在一个右连续的确定性马尔可夫策略，该策略依赖于当前状态和嵌入参数.考虑目标集为一个的马尔可夫决策过程，其中准则函数是最小函数的期望。我们制定的问题作为一个无限的地平线的情况下，经常性的类。我们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。并给出了一种策略改进方法.我们认为 ...更多信息 r一个随机最短路径问题与关联准则，其中对于图的每个节点，我们选择一个概率分布在一组后继节点，以便最优地到达给定的目标节点。我们制定这样的问题作为一个关联马尔可夫决策过程。我们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。给出了一种价值迭代方法和策略改进方法.我们考虑效用约束马尔可夫决策过程。在多个期望效用约束条件下，最大化总折扣奖励的期望效用。通过引入相应的拉格朗日函数，导出了效用约束优化问题的鞍点定理。约束最优策略的存在性由最优行动集和参数效用来刻画.我们考虑一个不等式条件，其中一边大于或等于另一边的倍数，当且仅当一个值是另一个变量的倍数时，等式成立。我们证明了一元函数的四对Golden不等式之间的交叉对偶性。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