Optimal investment with insurable background risk and nonlinear portfolio allocation frictions

We study optimal investment and insurance demand in a continuous-time model that combines risky assets with an insurable background risk. This risk takes the form of a jump-diffusion process that reduces the return rate of the agent's wealth. The main distinctive feature of our model is that the agent's decision on portfolio choice and insurance demand causes nonlinear friction in the dynamics of the wealth process. We use the HJB equation to find the optimal conditions for the agent to fully, partially, or totally insure against the background risk. We consider different types of friction, such as differential borrowing and lending rates. We also show a mutual-fund separation result and provide numerical examples.