Robust non-zero-sum stochastic differential game of two insurers with common shock and CDS transaction

Abstract

This paper considers the non-zero-sum stochastic differential game problem between two ambiguity-averse insurers (AAIs) with common shock. Each AAI’s surplus process consists of a proportional reinsurance protection and an investment in a money account, a stock and a credit default swap (CDS) with the objective of maximizing the expected utility of her relative terminal surplus with respect to that of her competitors. We consider default contagion risk of CDSs through a Markovian model with interacting default intensities. It is worthwhile to consider the uncertainty of the model on both the insurer herself and her competitors. In our model, we describe the surplus processes of two insurers by two jump-diffusion models with a common shock. Under jump-diffusion models, the robust Nash equilibrium strategies and the value functions for the all-default, one-default and all-alive case are derived under a worst-case scenario, respectively. Finally, through some numerical examples, we found some interesting results about the effects of some model parameters on the robust Nash equilibrium strategies, such as, the common shocks and the individual claims have the opposite effect on reinsurance investment.