Robust non-zero-sum stochastic differential game of two insurers with common shock and CDS transaction (Q6594800)

The authors of the paper consider the optimal reinsurance and investment problems under the objective of maximizing the expected utility in the fields of insurance and control theory. It is supposed that insurers can increase their returns by investing in risk-free and risky assets and buying reinsurance to transfer part of their risks. Mathematically, such a situation is described as the non-zero-sum stochastic differential game problem between two ambiguity-averse insurers with common shock. The authors suppose that the surplus process \( {U_i(t)},i\in{1,2},\) of the \(i\)-th insurer is described by a jump-diffusion risk model. \N\[\N\mathrm{d}U_i(t)=c_i\mathrm{d}t+\sigma_i\mathrm{d}B_i(t)-\mathrm{d}\bigg(\sum_{j=1}^{N_i(t)+N(t)}Y_{ij}\bigg), t\geqslant 0. \N\]\NHere \(c_i>0\) is the premium rate, \(\sigma_i\geqslant0\) is a constant, \(B_i(t)\) is a standard Brownian motion, which describes an additional source of uncertainty in the surplus process of the insurer, \N\[\N \sum_{j=1}^{N_i(t)+N(t)}Y_{ij} \N\]\Nis a compound Poisson process, which represents the aggregate claims of the \(i\)-th insurer up to time \(t\). The claim arrival process of the \(i\)-th insurer is homogeneous Poisson process \(N_i(t)\) and the common claim arrival process of the \(i\)-th insurer and her competitor is the another homogeneous Poisson process \(N(t)\). The individual claim sizes \(\{Y_{i1},Y_{i2},\dots\}\) are supposed to be i.i.d. positive random variables.\N\NAfter choosing the appropriate constraints for the model elements, 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. Through some numerical examples, the authors of the article present some results about the effects of the model parameters on the robust Nash equilibrium strategies.