Optimal Reinsurance and Investment Strategies under Strategic Interaction and Model Uncertainty

The study of investment and reinsurance optimization problems has developed as a time-honored research area, which has drawn significant interests both in the academia, the insurance and financial sectors. The present study discusses several reinsurance and investment optimization problems in a dynamic environment from two perspectives: (1) application of a robust approach to elucidate how market participants' attitudes towards ambiguity would impact the optimal decision rules; and (2) construction of analytical frameworks to investigate the effects of strategic interactions between two different decision-makers on their optimal strategies.

From the first perspective, considering that the findings yielded by extant experimental studies confirm that the decision-makers are not only risk-averse but also ambiguity-averse, in the current investigation, the focus is on the optimization problems in the presence of model uncertainty. It is further assumed that the reference model available to the decision-makers is an approximating model which may contain a specification error. Specifically, the financial and insurance models are characterized mathematically by the uncertainty about certain parameters in a diffusion model and a jump-type model. Moreover, the decision-makers obtain decision rules that are robust to model misspecification by maximizing their performance functionals over the worst-case scenario across a family of plausible models, which formulates max-min problems. The discrepancy between the alternative models and the reference model is defined through the concept of relative entropy, which serves as a penalty function in the optimization procedure.

Regarding the second perspective, we proceed along two directions. First, we formulate a competition between two insurance companies whose control policies impact those of their competitors due to their relative performance concerns and the correlation between their surplus processes. It is further assumed that they have access to the same financial market and their aggregated claims are governed by a common Poisson process describing the systematic insurance risk. The concept of relative performance concerns is incorporated into robust optimization problems under the expected utility maximization and mean-variance principle criteria. The system comprising the robust optimization problems pertaining to these two companies constitutes a robust non-zero-sum stochastic differential two-player game, for which the explicit expressions for Nash equilibrium strategies can be derived. Second, the bargaining between an insurer and a reinsurer, both of whom are ambiguity-averse, negotiating a reinsurance contract is studied. These two parties in a reinsurance policy form a principal-agent framework, which is essentially a Stackelberg game between the two contracting parties, allowing the insurer's and reinsurer's interests to be combined in a continuous-time setting. Under this framework, the reinsurer is regarded as a principal while the insurer is assigned the role of an agent, allowing the former to adjust the reinsurance premium according to the latter's reinsurance demands. We seek for the equilibrium optimal reinsurance arrangement by a two-step method and discuss the utility loss of the insurer and the reinsurer if they ignore model misspecification.

Finally, based on the study findings, implications for theory and practice are delineated before offering some suggestions for further research in this field.