Studies on Dynamic Optimization of Stochastic Systems with Multiple Criteria

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First, I examined an optimal stopping problem for a geometric Brownian motion with Poissonian jumps. Although it has been argued that so called smooth pasting technique (see Dixit (1993), and Dixit and Pindyck (1994)) is useful for such stochastic optimization problems, it seems that its mathematical validity is not sufficiently discussed so far. In this project, by taking a martingale approach, I showed that it is indeed mathematically valid under a set of some mild conditions on the parameters of the problem.Stochastic dominances (stochastic orders) and inequalities are very useful tools in various areas of economics and finance. The second purpose of this project was to describe main results obtained so far by using the idea of stochastic dominances in financial optimization. Especially, the emphasis is placed on the demand and shift effect problems in portfolio selection. Some other examples, which are not related directly to optimization problems, are also gathered to demonstrate the wide spectrum of application areas of stochastic dominances in finance. Further, since several stochastic dominances and related inequalities which are known in the reliability and maintainability theory, are very useful even in finance theory, the next purpose of this project was to provide a brief survey of the useful known results concerning stochastic orders and their applications developed in various areas of the reliability and maintainability theory.