Optimal Investment, Consumption, and Insurance with Durable Goods under Stochastic Depreciation Risk

We study an infinite-horizon optimal investment, consumption and insurance problem for an economic agent who consumes a perishable and a durable good. The agent trades in a risk-free asset, a risky asset, and a durable good whose price follows a correlated diffusion, while the stock of the durable good depreciates deterministically and is subject to insurable Poisson loss shocks. The agent can partially hedge these shocks via an insurance contract with loading and chooses optimal perishable consumption, portfolio holdings, and insurance coverage to maximise expected discounted CRRA utility. Exploiting the homogeneity of the problem, we reduce the Hamilton--Jacobi--Bellman equation to a static one-dimensional optimisation over constant portfolio shares and derive a semi-explicit optimal strategy. We then prove a verification theorem for the associated jump-diffusion wealth process with insurance, establishing the existence and optimality of this constant-fraction strategy under explicit transversality conditions for both risk-aversion regimes $0<\gamma<1$ and $\gamma>1$. Numerical experiments illustrate the impact of stochastic depreciation risk and insurance loading on the optimal allocation to financial assets, durable goods, and insurance coverage.