• The Pure and Applied Mathematics
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• v.22 no.4
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• pp.333-342
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• 2015

We show that the set of priors in the representation of Choquet expectation is the one of equivalent martingale measures under some conditions, when given capacity is submodular. It is proven via Peng’s g-expectation and related topics.

• The Pure and Applied Mathematics
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• v.20 no.4
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• pp.287-298
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• 2013

A standard deviation has been a starting point for a mathematical definition of risk. As a remedy for drawbacks such as subadditivity property discouraging the diversification, coherent and convex risk measures are introduced in an axiomatic approach. Choquet expectation and g-expectations, which generalize mathematical expectations, are widely used in hedging and pricing contingent claims in incomplete markets. The each risk measure or expectation give rise to its own pricing rules. In this paper we investigate relationships among dynamic risk measures, Choquet expectation and dynamic g-expectations in the framework of the continuous-time asset pricing.

• Proceedings of the Korean Society of Computational and Applied Mathematics Conference
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• 2003.09a
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• pp.7-7
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• 2003

We studied closed set-valued Choquet integrals in two papers(1997, 2000) and convergence theorems under some sufficient conditions in two papers(2003), for examples : (i) convergence theorems for monotone convergent sequences of Choquet integrably bounded closed set-valued functions, (ii) covergence theorems for the upper limit and the lower limit of a sequence of Choquet integrably bounded closed set-valued functions. In this presentation, we consider fuzzy number-valued functions and define Choquet integrals of fuzzy number-valued functions. But these concepts of fuzzy number-valued Choquet inetgrals are all based on the corresponding results of interval-valued Choquet integrals. We also discuss their properties which are positively homogeneous and monotonicity of fuzzy number-valued Choquet integrals. Furthermore, we will prove convergence theorems for fuzzy number-valued Choquet integrals. They will be used in the following applications : (1) Subjectively probability and expectation utility without additivity associated with fuzzy events as in Choquet integrable fuzzy number-valued functions, (2) Capacity measure which are presented by comonotonically additive fuzzy number-valued functionals, and (3) Ambiguity measure related with fuzzy number-valued fuzzy inference.

• The Pure and Applied Mathematics
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• v.23 no.4
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• pp.339-345
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• 2016

The set of priors in the representation of Choquet expectation is expressed as the one of equivalent martingale measures under some conditions. We show that the set of priors, $\mathcal{Q}_c$ in (1.1) is the same set of $\mathcal{Q}^{\theta}$ in (1.3).

• The Pure and Applied Mathematics
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• v.23 no.4
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• pp.377-383
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• 2016

The set of priors in the representation of coherent risk measure is expressed in terms of quantile function and increasing concave function. We show that the set of prior, $\mathcal{Q}_c$ in (1.2) is equal to the set of $\mathcal{Q}_m$ in (1.6), as maximal representing set $\mathcal{Q}_{max}$ defined in (1.7).

• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 2003.09a
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• pp.28-31
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• 2003

In the present work, a main purpose is to propose a fuzzy integral-based aggregation framework to complementarily combine partial information due to lack of completeness. Based on Choquet integral (CI) viewed as monotone expectation, we take into account complementary, non-interactive, and substitutive aggregations of different sources of defective information. A CI-based system representing upper, conventional, and lower expectations is designed far handling three aggregation attitudes towards uncertain information. In particular, based on Choquet integrals for belief measure, probability measure, and plausibility measure, CI$\_$bi/-, CI$\_$pr/ and CI$\_$pl/-aggregator are constructed, respectively. To illustrate a validity of proposed aggregation framework, multiple matching systems are developed by combining three simple individual template-matching systems and tested under various image variations. Finally, compared to individual matchers as well as other traditional multiple matchers in terms of an accuracy rate, it is shown that a proposed CI-aggregator system, {CI$\_$bl/-aggregator, CI$\_$pl/-aggregator, Cl$\_$pl/-aggregator}, is likely to offer a potential framework for either enhancing completeness or for resolving conflict or for reducing uncertainty of partial information.