Non-Additive Criterion on Controlled Markov Chains and its Applications to Mathematical Finance

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This project studies from a viewpoint of non-additivity, which is well-known as non-linearity in criterion. Our aim is to clarify an optimal structure both in policy and in system dynamics under the expected utility criteria. Further we also apply the results we have obtained for optimization scheme to non-optimization problems such as option evaluation in mathematical finance.It has been well known that dynamic optimizations such as in Markov decision processes have treated additive (linear) criteria e.g. discounted total expected reward. These optimizations are relatively easily performed because of linearity in expectation operator: However what will happen when we optimize such non-additive criteria? Our study begins with abandoning the linearity in expectation. Instead, we focus our attention to (a)monotonicity in expectation operator, (b)associativity in reward criteria and (c)successive applicability of state dynamics.By using these three properties, we have succeeded in establi … More shing a large variety of optimization methods and in recognizing that these dynamic methods turn out to be fruitful in implemnentation/calculation. To be more concretely specific, we have obtained the following four results.(1)To have introduced policy classes and classified them into Markov, general, primitive and expanded Markov,(2)To have introduced two criterion classes and classified both into simple criterion (additive, multiplicative, maximum and terminal) and compound criterion (range, variance, ratio, sum excluding extrema and mid-range),(3)To have associated the policy classes with the two criterion classes and presented how to imbed the original problem into an expanded class of problems,(4)To have applied the results obtained for wide class of optimization problems in (1)-(3) to a class of non-optimization problems in mathematical finance such as option pricing.Throughout this study, we have clarified that a large class of optimization methods have been useful for a wide variety of criteria. These methods have been restricted to a small class of deterministic problems. Now our approach has shown that these methods are applicable to a huge class of optimization and non-optimization problems in stochastic, fuzzy or non-deterministic problems. In particular we have developed an evaluation method of a large class of options (derivatives) in mathematical finance. This is called dynamic pricing or recursive evaluation. Further more we have succeeded in developing some new derivatives such as options with random expiration date…Pacific options, look back American options and others.As a summary we have made dynamic optimization method fiuitful Thus dynamic programming, recursive method, and invariant imbedding turn out to be useful for a wide class of optimization and/or evaluation problems which inherits (i)non-additivity, (ii)non-linearity or (iii)non-determinancy. This project has just opened the door to overcome the three difficulties. Less