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An optimal insurance design problem under Knightian uncertainty. (English) Zbl 1277.91075

The paper investigates an insurance market with Knightian uncertainty; in particular it focuses on the optimal design problem, where both the insurer and the insured have ambiguity on loss distribution.
At first, the optimal insurance contract is obtained for the insured, supposing that the insurer perfectly knows the loss variable and is risk neutral. Then the result is extended to the case in which both the insurer and the insured have Knightian uncertainties.
Finally, some special cases are treated in order to illustrate the optimal indemnity.

MSC:

91B30 Risk theory, insurance (MSC2010)
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[1] Alary, D., Gollier, C., Treich, N.: Risk, ambiguity and insurance. Working paper presented at the World Risk and Insurance Congress in Singapore (2010)
[2] Arrow K.: Essays in the Theory of Risk Bearing. Markham, Chicago (1971) · Zbl 0215.58602
[3] Bernard, C.; Tian, W., Optimal reinsurance arrangements under tail risk measures, J. Risk Insur., 76, 709-725, (2009)
[4] Bernard, C.; Tian, W., Optimal insurance policies when insurers implement risk management metrics, Geneva Risk Insur. Rev., 35, 47-80, (2010)
[5] Bewley, T.F., Knightian decision theory, part I, Decis. Econ. Finance, 25, 79-110, (2002) · Zbl 1041.91023
[6] Billot, A.; Chateauneuf, A.; Gilboa, I.; Tallon, J.-M., Sharing beliefs: between agreeing and disagreeing, Econometrica, 68, 685-694, (2000) · Zbl 1023.91503
[7] Borch, K., Equilibrium in a reinsurance market, Econometrica, 30, 424-444, (1962) · Zbl 0119.36504
[8] Breuer, M., Multiple losses, ex ante moral hazard and the implications for umbrella policies, J. Risk Insur., 72, 525-538, (2005)
[9] Cabantous, L., Ambiguity aversion in the field of insurance: insurer’s attitude to imprecise and conflicting probability estimates, Theory Decis., 62, 219-240, (2007) · Zbl 1121.91056
[10] Carlier, G.; Dana, R.-A., Two-persons efficient risk-sharing and equilibria for concave law-invariant utilities, Econ. Theory, 36, 189-223, (2008) · Zbl 1152.91035
[11] Carlier, G.; Dana, R.-A.; Shahidi, N., Efficient insurance contracts under epsilon-contaminated utilities, Geneva Papers Risk Insur. Theory, 28, 59-71, (2003)
[12] Chateauneuf, A.; Dana, R.-A.; Tallon, J.-M., Optimal risk sharing rules and equilibria with Choquet expected utilities, J. Math. Econ., 34, 191-214, (2000) · Zbl 1161.91434
[13] Chen, Z.; Epstein, L., Ambiguity, risk and asset returns in continuous time, Econometrica, 70, 1403-1443, (2002) · Zbl 1121.91359
[14] Cooper, R.; Hayes, B., Multi-period insurance contracts, Int. J. Ind. Organ., 5, 211-231, (1987)
[15] Crocker, K.; Snow, A., The efficiency of competitive equilibria in insurance markets with asymmetric information, J. Public Econ., 26, 207-219, (1985)
[16] Cummins, J.; Mahul, O., Optimal insurance with divergent beliefs about insurer total default risk, J. Risk Uncertain., 27, 121-138, (2003) · Zbl 1054.91050
[17] Cummins, J.D.; Lalonde, D.; Phillips, R., The basis risk of index linked catastrophe loss securities, J. Financial Econ., 77, 77-111, (2004)
[18] Doherty, N.; Schlesinger, H., Optimal insurance in incomplete markets, J. Political Econ., 91, 1045-1054, (1983)
[19] Doherty, N.; Schlesinger, H., The optimal deductible for an insurance policy when insurance wealth is random, J. Bus., 56, 555-565, (1983)
[20] Doherty, N.A.; Eeckhoudt, L., Optimal insurance without expected utility: the dual theory and the linearity of insurance contracts, J. Risk Uncertain., 10, 157-179, (1995) · Zbl 0847.90036
[21] Duffie, D.; Epstein, L., Stochastic differential utility, Econometrica, 60, 353-394, (1992) · Zbl 0763.90005
[22] El Karoui, N.; Peng, S.; Quenez, M., Backward stochastic differential equations in finance, Math. Finance, 7, 1-71, (1997) · Zbl 0884.90035
[23] El Karoui, N.; Peng, S.; Quenez, M., A dynamic maximum principle for the optimization of recursive utilities under constraints, Annal. Appl. Probab., 11, 64-694, (2001) · Zbl 1040.91038
[24] Ellsberg, D., Risk, ambiguity, and the savage axioms, Q. J. Econ., 75, 643-669, (1961) · Zbl 1280.91045
[25] Epstein, L.; Schneider, M., Recursive multiple-priors, J. Econ. Theory, 113, 1-33, (2003) · Zbl 1107.91360
[26] Froot, K., The market for catastrophe exchange: a clinic exam, J. Financial Econ., 60, 529-571, (2001)
[27] Gilboa, I.; Schmeidler, D., Maximum expected utility with a non-unique prior, J. Math. Econ., 18, 141-153, (1989) · Zbl 0675.90012
[28] Gollier, C., Pareto-optimal risk sharing with fixed costs per claim, Scand. Actuar. J., 13, 62-73, (1987) · Zbl 0633.62104
[29] Gollier, C.; Schlesinger, H., Second-best insurance contract design in an incomplete market, Scand. J. Econ., 97, 123-135, (1995) · Zbl 0910.90107
[30] Gollier, C.; Schlesinger, H., Arrow’s theorem on the optimality of deductibles: a stochastic dominance approach, Econ. Theory, 7, 359-363, (1996) · Zbl 0852.90047
[31] Golubin, A., Pareto-optimal insurance policies in the models with a premium based on the actuarial value, J. Risk Insur., 73, 469-487, (2006)
[32] Ho, J.; Keller, L.; Kunreuther, H.; etal., Risk, ambiguity and insurance, J. Risk Uncertain., 2, 5-35, (1989)
[33] Hogarth, R.; Kunreuther, H., Ambiguity and insurance decisions, Am. Econ. Rev., 75, 386-390, (1985)
[34] Hogarth, R.; Kunreuther, H., Risk, ambiguity and insurance, J. Risk Uncertain., 2, 5-35, (1989)
[35] Ji, S.; Peng, S., Terminal perturbation method for the backward approach to continuous-time Mean-variance portfolio selection, Stoch. Process. Appl., 118, 952-967, (2008) · Zbl 1152.60051
[36] Ji, S.; Zhou, X., A generalized Neyman-person lemma for g-probabilities, Probab. Theory Rel. Fields, 148, 645-669, (2010) · Zbl 1197.93163
[37] Johnson, E.; Hershey, J.; Meszaros, J.; Kunreuther, H., Framing, probability distortions and insurance decisions, J. Risk Uncertain., 7, 35-51, (1993)
[38] Knight F.: Risk, Uncertainty and Profit. Houghton Mifflin Company, Boston (1921)
[39] Kunreuther, H.; Hogarth, R.; Meszaros, J., Insurer ambiguity and market failure, J. Risk Uncertain., 7, 71-87, (1993)
[40] Landsberger, M.; Meilijson, I., General model of insurance under adverse selection, Econ. Theory, 14, 331-352, (1999) · Zbl 0934.91034
[41] Luenberger D.: Optimization by Vector Space Methods. Wiley, New York (1969) · Zbl 0176.12701
[42] Ma, J.; Protter, P.; Martin, J.S.; Torres, S., Numerical methods for backward stochastic differential equations, Annal. Appl. Probab., 12, 302-316, (2002) · Zbl 1017.60074
[43] Mukerji, S.; Tallon, J.-M., Ambiguity aversion and incompleteness of financial markets, Rev. Econ. Stud., 68, 883-904, (2001) · Zbl 1051.91047
[44] Nishimura, K.; Ozaki, H., Search and Kinghtian uncertainty, J. Econ. Theory, 119, 299-333, (2004) · Zbl 1099.91070
[45] Nishimura, K.; Ozaki, H., Irereversible investment and Kinghtian uncertainty, J. Econ. Theory, 136, 668-694, (2007) · Zbl 1281.91061
[46] Obstfeld, M., Risk-taking, global diversification, and growth, Am. Econ. Rev., 84, 1310-1329, (1994)
[47] Peng, S.; El Karoui, N. (ed.); Mazliak Esses, L. (ed.), BSDE and related g-expectation, 141-159, (1997), Reading · Zbl 0892.60066
[48] Raviv, A., The design of an optimal insurance policy, Am. Econ. Rev., 69, 84-96, (1979)
[49] Rigotti, L.; Shannon, C.; Strzalecki, T., Subjective beliefs and ex ante trade, Econometrica, 76, 1167-1190, (2008) · Zbl 1152.91655
[50] Rothschild, M.; Stiglitz, J.E., Equilibrium in competitive insurance markerts: an essay on the economics of imperfect information, Q. J. Econ., 90, 629-649, (1976)
[51] Schmeidler, D., Subjective probability and expected utility without additivity, Econometrica, 57, 571-587, (1989) · Zbl 0672.90011
[52] Uppal, R.; Wang, T., Misspecification and under-diversification, J. Finance, 58, 2465-2486, (2003)
[53] Vazquez, F.; Watt, R., A theorem on optimal multi-period insurance contracts with commitment, Insur. Math. Econ., 24, 273-280, (1999) · Zbl 0945.62114
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