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The optimization problem is split into two minimization problems using an infimum representation for spectral risk measures. The MDP is assumed to have Borel state and action spaces and the cost function may be unbounded above. We study the minimization of a spectral risk measure of the total discounted cost generated by a Markov Decision Process (MDP) over a finite or infinite planning horizon. We also investigate whether or not our axioms have forms of representation. We explore whether useful methods of measuring portfolio diversification satisfy our axioms. We provide the decision-theoretic foundations of our axioms by studying their compatibility with investors' preference for diversification in two important decision theories under risk: the expected utility theory and Yaari's dual theory. We propose a set of nine desirable axioms for this class of diversification measures, and name the measures satisfying these axioms coherent diversification measures that we distinguish from the notion of coherent risk measures. We offer the first step towards a rigorous theory of correlation diversification measures. Four categories of portfolio diversification measures can be distinguished: the law of large numbers diversification measures, the correlation diversification measures, the market portfolio diversification measures and the risk contribution diversification measures. This paper provides an axiomatic foundation of the measurement of diversification in a one-period portfolio theory under the assumption that the investor has complete information about the joint distribution of asset returns. As an application, we present a dynamic extension of the classical static optimal reinsurance problem, where an insurance company minimizes its cost of capital. Our results include the findings in B\"auerle and Ott (2011) in the special case that the risk measure is Expected Shortfall. Regarding the infinite dimensional outer minimization problem, we prove the existence of a solution and derive an algorithm for its numerical approximation. We show that the inner minimization problem can be solved as an ordinary MDP on an extended state space and give sufficient conditions under which an optimal policy exists.
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Furthermore, we derive formulas for truncated elliptical models of losses and provide formulas for selected members of such models. We then investigate the fundamental properties of the proposed measure and show its unique features and implications in the risk measurement process. In this paper, we propose a natural generalization of the Certainty Equivalent measure that, similar to the Value at Risk measure, is focusing on the tail distribution of the risk, and thus, focusing on extreme financial and insurance risk events. One of the most extensively used risk measures is the Value at Risk, which is investigated and used by both researchers and practitioners as a powerful tool that measures the risk under some quantile level which allows focusing on the extreme amount of losses. Therefore, it plays an essential role in utility-based decision making. Certainty Equivalent is a utility-based measure that performs as a measure in which investors are indifferent between this measure and investment that holds some uncertainty.