# Objective

This course will deal with the state-of-the-art in the theory of Pareto-optimal (re)insurance design under model uncertainty and/or non-Expected-Utility preferences, as well as provide an introduction to Pareto optimality in problems of peer-to-peer collaborative insurance. Specifically:

- We will start with some background material on the theory of decision-making under uncertainty, from the classical work on Expected-Utility Theory (EUT) of von Neumann and Morgenstern, Savage, and De Finetti, to the more recent work on ambiguity and probability distortions (Quiggin, Yaari, Schmeidler, Gilboa, Amarante, Maccheroni-Marinacci).
- We will cover some required mathematical tools, such as non-additive measure theory, probability distortions, Choquet integration, the theory of equimeasurable rearrangements, as well as risk measures, their properties, and their representations. We will pay special attention to distortion risk measures and spectral risk measures.
- We will then formally introduce a general model of the insurance market, following the work of Carlier and Dana [8], and study the existence and characterization of Pareto optima in this general setup. As a special case, we will consider the classical formulation of the optimal (re)insurance problem due to Arrow, in the framework of EUT, as well as more recent work extending Arrow's setting to situations of belief heterogeneity between the two agents.
- We will then proceed to formulating several problems that extend the insurance model above to more general setting with non-EUT preferences, distortion risk measures, or situations of model uncertainty.
- Finally, we will go over a brief introduction to conditional mean risk sharing and peer-to-peer collaborative insurance.

# Planning

**Introduction to Choice under Uncertainty**

General overview of decision theory

Models of decision-making under uncertainty**Pareto-Optimal Insurance Contracts under EUT**

Some mathematical tools

General problem formulation

Pareto optimality

The special case of Arrow's setting

Pareto optima and comonotonicity

Pareto optima under belief heterogeneity**Non-Additive Measures and Choquet Integration**

Capacities and their properties

The Choquet integral and its properties

Mathematics of the RDEU model

Distortion risk measures and spectral risk measures**Pareto-Optimal Insurance Contracts under non-EUT Models**

Problem formulation

The case of RDEU

The case of Choquet EU

The case of MEU

A note on belief heterogeneity under non-EUT**Pareto-Optimality in Reinsurance Markets**

General problem formulation

Pareto optimality with translation invariant preferences

Distortion risk measures and the marginal indemnification approach**Peer-to-Peer Risk Sharing Mechanisms: An Introduction**

General problem formulation

Pareto-optimality, convex ordering, and conditional mean risk sharing

# Références

[1] K.J. Arrow. Essays in the Theory of Risk-Bearing. Chicago: Markham Publishing Company, 1971.

[2] V.A. Asimit and T.J. Boonen. Insurance with Multiple Insurers: a Game-Theoretic Approach. European Journal of Operational Research, 267(2):778{790, 2018.

[3] H. Assa. On Optimal Reinsurance Policy with Distortion Risk Measures and Premiums. Insurance: Mathematics and Economics, 61:70{75, 2015.

[4] C. Bernard, X. He, J.A. Yan, and X.Y. Zhou. Oprimal Insurance Design under Rank-Dependent Expected Utility. Mathematical Finance, 25(1):154{186, 2015.

[5] C. Birghila, M. Ghossoub, and T. Boonen. Optimal Insurance under Maxmin Expected Utility. Working Paper, 2020.

[6] T.J. Boonen and M. Ghossoub. On the Existence of a Representative Reinsurer under Heterogeneous Beliefs. Insurance Mathematics and Economics, 88:209{225, 2019.

[7] T.J. Boonen and M. Ghossoub. Optimal Reinsurance with Multiple Reinsurers: Distortion Risk Measures, Distortion Premium Principles, and Heterogeneous Beliefs. Insurance Mathematics and Economics, 2021. Forthcoming.

[8] G. Carlier and R.A. Dana. Pareto Ecient Insurance Contracts when the Insurer's Cost Function is Discontinuous. Economic Theory, 21(4):871{893, 2003.

[9] G. Carlier and R.A. Dana. Rearrangement Inequalities in Non-convex Insurance Models.Journal of Mathematical Economics, 41(4-5):483{503, 2005.

[10] G. Carlier, R.A. Dana, and A. Galichon. Pareto Eciency for the Concave Order and Multivariate Comonotonicity. Journal of Economic Theory, 147(1):207{229, 2012.

[11] Y. Chi. On the Optimality of a Straight Deductible under Belief Heterogeneity. ASTIN Bulletin, 49(1):243{262, 2019.

[12] K.M. Chong and N.M. Rice. Equimeasurable Rearrangements of Functions. Queens Papers in Pure and Applied Mathematics, 28, 1971.

[13] D. Denneberg. Non-Additive Measure and Integral. Kluwer Academic Publishers, 1994.3

[14] M. Denuit. Investing in Your Own and Peers' Risks: the Simple Analytics of P2P Insurance. European Actuarial Journal, 10(2):335{359, 2020.

[15] M. Denuit and J. Dhaene. Convex Order and Comonotonic Conditional Mean Risk Sharing. Insurance Mathematics and Economics, 51(2):265{270, 2012.

[16] M. Denuit, J. Dhaene, M. Goovaerts, and R. Kaas. Actuarial Theory for Dependent Risks: Measures, Orders and Models. John Wiley & Sons, 2005.

[17] M. Denuit and C.Y. Robert. Risk Sharing under the Dominant Peer-to-Peer Property and Casualty Insurance Business Models. Risk Management and Insurance Review, 24(2):1{25, 2021.

[18] H. Follmer and A. Schied. Stochastic Finance: An Introduction in Discrete Time { 4th ed. Walter de Gruyter, 2016.

[19] M. Ghossoub. Budget-Constrained Optimal Insurance With Belief Heterogeneity. Insurance Mathematics and Economics, 89:79{91, 2019.

[20] M. Ghossoub. Optimal Insurance under Rank-Dependent Expected Utility. Insurance Mathematics and Economics, 87:51{66, 2019.

[21] I. Gilboa. Theory of Decision under Uncertainty. Cambridge University Press, 2009.

[22] I. Gilboa and M. Marinacci. Ambiguity and the Bayesian Paradigm. In D. Acemoglu, M. Arellano, and E. Dekel (eds.), Advances in Economics and Econometrics: Theory and Applications, Tenth World Congress of the Econometric Society. Cambridge University Press, 2013.

[23] M. Marinacci and L. Montrucchio. Introduction to the Mathematics of Ambiguity. In I. Gilboa (ed.), Uncertainty in Economic Theory: Essays in Honor of David Schmeidler's 65th Birthday, pages 46{107. Routledge, London, 2004.

[24] M. Shaked and J.G. Shanthikumar. Stochastic Orders. Springer, 2007.

[25] Z.Q. Xu, X.Y. Zhou, and S.C. Zhuang. Optimal Insurance under Rank-Dependent Utility and Incentive Compatibility. Mathematical Finance, 21(4):503{34, 2018.4

[2] V.A. Asimit and T.J. Boonen. Insurance with Multiple Insurers: a Game-Theoretic Approach. European Journal of Operational Research, 267(2):778{790, 2018.

[3] H. Assa. On Optimal Reinsurance Policy with Distortion Risk Measures and Premiums. Insurance: Mathematics and Economics, 61:70{75, 2015.

[4] C. Bernard, X. He, J.A. Yan, and X.Y. Zhou. Oprimal Insurance Design under Rank-Dependent Expected Utility. Mathematical Finance, 25(1):154{186, 2015.

[5] C. Birghila, M. Ghossoub, and T. Boonen. Optimal Insurance under Maxmin Expected Utility. Working Paper, 2020.

[6] T.J. Boonen and M. Ghossoub. On the Existence of a Representative Reinsurer under Heterogeneous Beliefs. Insurance Mathematics and Economics, 88:209{225, 2019.

[7] T.J. Boonen and M. Ghossoub. Optimal Reinsurance with Multiple Reinsurers: Distortion Risk Measures, Distortion Premium Principles, and Heterogeneous Beliefs. Insurance Mathematics and Economics, 2021. Forthcoming.

[8] G. Carlier and R.A. Dana. Pareto Ecient Insurance Contracts when the Insurer's Cost Function is Discontinuous. Economic Theory, 21(4):871{893, 2003.

[9] G. Carlier and R.A. Dana. Rearrangement Inequalities in Non-convex Insurance Models.Journal of Mathematical Economics, 41(4-5):483{503, 2005.

[10] G. Carlier, R.A. Dana, and A. Galichon. Pareto Eciency for the Concave Order and Multivariate Comonotonicity. Journal of Economic Theory, 147(1):207{229, 2012.

[11] Y. Chi. On the Optimality of a Straight Deductible under Belief Heterogeneity. ASTIN Bulletin, 49(1):243{262, 2019.

[12] K.M. Chong and N.M. Rice. Equimeasurable Rearrangements of Functions. Queens Papers in Pure and Applied Mathematics, 28, 1971.

[13] D. Denneberg. Non-Additive Measure and Integral. Kluwer Academic Publishers, 1994.3

[14] M. Denuit. Investing in Your Own and Peers' Risks: the Simple Analytics of P2P Insurance. European Actuarial Journal, 10(2):335{359, 2020.

[15] M. Denuit and J. Dhaene. Convex Order and Comonotonic Conditional Mean Risk Sharing. Insurance Mathematics and Economics, 51(2):265{270, 2012.

[16] M. Denuit, J. Dhaene, M. Goovaerts, and R. Kaas. Actuarial Theory for Dependent Risks: Measures, Orders and Models. John Wiley & Sons, 2005.

[17] M. Denuit and C.Y. Robert. Risk Sharing under the Dominant Peer-to-Peer Property and Casualty Insurance Business Models. Risk Management and Insurance Review, 24(2):1{25, 2021.

[18] H. Follmer and A. Schied. Stochastic Finance: An Introduction in Discrete Time { 4th ed. Walter de Gruyter, 2016.

[19] M. Ghossoub. Budget-Constrained Optimal Insurance With Belief Heterogeneity. Insurance Mathematics and Economics, 89:79{91, 2019.

[20] M. Ghossoub. Optimal Insurance under Rank-Dependent Expected Utility. Insurance Mathematics and Economics, 87:51{66, 2019.

[21] I. Gilboa. Theory of Decision under Uncertainty. Cambridge University Press, 2009.

[22] I. Gilboa and M. Marinacci. Ambiguity and the Bayesian Paradigm. In D. Acemoglu, M. Arellano, and E. Dekel (eds.), Advances in Economics and Econometrics: Theory and Applications, Tenth World Congress of the Econometric Society. Cambridge University Press, 2013.

[23] M. Marinacci and L. Montrucchio. Introduction to the Mathematics of Ambiguity. In I. Gilboa (ed.), Uncertainty in Economic Theory: Essays in Honor of David Schmeidler's 65th Birthday, pages 46{107. Routledge, London, 2004.

[24] M. Shaked and J.G. Shanthikumar. Stochastic Orders. Springer, 2007.

[25] Z.Q. Xu, X.Y. Zhou, and S.C. Zhuang. Optimal Insurance under Rank-Dependent Utility and Incentive Compatibility. Mathematical Finance, 21(4):503{34, 2018.4