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

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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