We discuss an optimal excess-of-loss reinsurance contract in a continuous-time principal-agent framework where the surplus of the insurer (agent/he) is described by a classical Cramer-Lundberg (C-L) model. In addition to reinsurance, the insurer and the reinsurer (principal/she) are both allowed to invest their surpluses into a financial market containing one risk-free asset (e.g. a short-rate account) and one risky asset (e.g. a market index). In this paper, the insurer and the reinsurer are ambiguity averse and have specific modeling risk aversion preferences for the insurance claims (this relates to the jump term in the stochastic models) and the financial market’s risk (this encompasses the models’ diffusion term). The reinsurer designs a reinsurance contract that maximizes the exponential utility of her terminal wealth under a worst-case scenario which depends on the retention level of the insurer. By employing the dynamic programming approach, we derive the optimal robust reinsurance contract, and the value functions for the reinsurer and the insurer under this contract. In order to provide a more explicit reinsurance contract and to facilitate our quantitative analysis, we discuss the case when the claims follow an exponential distribution; it is then possible to show explicitly the impact of ambiguity aversion on the optimal reinsurance.

Frederi G. Viens is Professor and Chair of Michigan State University’s Department of Statistics and Probability. He started in August 2016. There, he directs the Actuarial Science program, and is interim acting director of Statistical Consulting. He was previously a Professor of Statistics and Mathematics at Purdue University (2000-2015); Program Director in the Division of Mathematical Sciences at the National Science Foundation in Arlington, VA (2015-2016); Co-Chair for Pure Mathematics for Canada’s National Science and Engineering Research Council (2011-2012); Science Advisor for the U.S. Assistant Secretary of State for African Affairs in Washington DC (2010-2011), and held positions in Spain, France, Texas, and Chile, in teaching and research roles.

Viens is the author of over 70 research papers on probability theory, stochastic analysis, and statistics for stochastic processes, with applications in finance, insurance, climatology, hydrology, agricultural economics, development economics, soil science, nuclear physics, and human health management. Some of his current emphases are to use stochastic analysis to study the detailed asymptotic properties of estimators for stochastic processes, use robust optimization to give investment and risk-management recommendations to insurance companies, and use computational Bayesian tools in a variety of applied problems. He continues to pursue a strong interest in scientific capacity development in Africa.

His distinctions include Fellow of the Institute of Mathematical Statistics (2012), Franklin Fellow of the U.S. State Department (2010), and Fulbright Research Scholar in Villetaneuse, France (2004). He is the Associate Editor of several prestigious journals, including the Annals of Finance, Annals of Probability, Bernoulli, the Electronic Journal of Statistics, and others. He is Founding Editor and Editor-in-Chief of the interdisciplinary data science journal High Frequency. He has organized numerous high-profile international conferences, both in the United States and overseas, including the Conference on High Frequency Finance (7 editions since 2009). He is routinely invited as sessional and plenary speaker at major international conferences in his field and other fields, including in finance, applied economics, nuclear physics, and agricultural sustainability. His research has been supported by the National Science Foundation since 2003, and recently by the Office of Naval Research, two US federal agencies.

Viens holds a Maîtrise de Mathématiques Pures from the Université de Paris 7, France (1991), and his Ph.D. in Mathematics from the University of California, Irvine (1996).