Optimal dividend strategies with reinsurance under contagious systemic risk

This paper studies the multidimensional mixed singular-regular stochastic control problems subject to reduced-form default driven by contagious intensities. The dynamic process of surplus is given by a system of diffusion processes with two controls, and the intensity of the reducedform model increases when defaults occur. We derive the recursive Hamilton-Jacobi-Bellman variational inequalities by the dynamic programming principle and present analytical and recursive solutions. We prove that the solutions are classical and recursively associated with each other by the default states. The verification theorem is presented.