Abstract
An optimal risk sharing problem for agents with utility functionals depending only on the expected value and a deviation measure of an uncertain payoff has been studied. The agents are assumed to have no initial endowments. A set of Pareto-optimal solutions to the problem has been characterized, and a particular solution from the set has been suggested. If an equilibrium exists, the suggested solution coincides with an equilibrium solution. As special cases, the optimal risk sharing problem in the form of expected gain maximization and the problem with a linear mean-deviation utility functional including averse and coherent risk measures have been addressed. In the case of expected gain maximization, the existence of an equilibrium has been shown.
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References
Aase, K. (2002). Perspectives of risk sharing. Scandinavian Actuarial Journal, 2, 73–128.
Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9, 203–227.
Borch, K. H. (1962). Equilibrium in a reinsurance market. Econometrica, 30, 424–44.
Burgert, C., & Ruschendorf, L. (2008). Allocation of risks and equilibrium in markets with finitely many traders. Insurance. Mathematics & Economics, 42(1), 177–188.
Dana, R.-A. (2005). A representation result for concave Schur-concave functions. Mathematical Finance, 15(4), 613–634.
Dana, R.-A., & Le Van, C. (2007). Overlapping sets of priors and the existence of efficient allocations and equilibria for risk measures. EconPapers.
Follmer, H., & Schied, A. (2004). Stochastic finance (2nd edn.). Berlin: de Gruyter.
Grechuk, B., Molyboha, A., & Zabarankin, M. (2008). Mean-deviation analysis in the theory of choice. Preprint.
Grechuk, B., Molyboha, A., & Zabarankin, M. (2009a). Cooperative games with general deviation measures. Under the second stage of review in Mathematical Finance.
Grechuk, B., Molyboha, A., & Zabarankin, M. (2009b). Maximum entropy principle with general deviation measures. Mathematics of Operations Research, 34(2), 445–467.
Jouini, E., Schachermayer, W., & Touzi, N. (2008). Optimal risk sharing for law invariant monetary utility functions. Mathematical Finance, 18(2), 269–292.
Kusuoka, S. (2001). On law invariant coherent risk measures. Advances in Mathematical Economics, 3, 83–95.
Landsberger, M., & Meilijson, I. (1994). Co-monotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion. Annals of Operation Research, 52, 97–106.
Rockafellar, R. T., Uryasev, S., & Zabarankin, M. (2002) Deviation measures in risk analysis and optimization (Report 2002-7). ISE Dept., Univ. of Florida.
Rockafellar, R. T., Uryasev, S., & Zabarankin, M. (2006a). Generalized deviations in risk analysis. Finance and Stochastics, 10(1), 51–74.
Rockafellar, R. T., Uryasev, S., & Zabarankin, M. (2006b). Optimality conditions in portfolio analysis with general deviation measures. Mathematical Programming, 108(2–3), 515–540.
Rockafellar, R. T., Uryasev, S., & Zabarankin, M. (2006c). Master funds in portfolio analysis with general deviation measures. The Journal of Banking and Finance, 30(2), 743–77.
Rockafellar, R. T., Uryasev, S., & Zabarankin, M. (2007). Equilibrium with investors using a diversity of deviation measures. The Journal of Banking and Finance, 31(11), 3251–3268.
Rockafellar, R. T., Uryasev, S., & Zabarankin, M. (2008). Risk tuning with generalized linear regression. Mathematics of Operations Research, 33(3), 712–729.
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Grechuk, B., Zabarankin, M. Optimal risk sharing with general deviation measures. Ann Oper Res 200, 9–21 (2012). https://doi.org/10.1007/s10479-010-0834-7
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DOI: https://doi.org/10.1007/s10479-010-0834-7