Optimal Investment with Time-Varying Stochastic Endowments

This paper considers a utility maximization and optimal asset allocation problem in the presence of a stochastic endowment that cannot be fully hedged through trading in the financial market. After studying continuity properties of the value function for general utility functions, we rely on the dynamic programming approach to solve the optimization problem for power utility investors including the empirically relevant and mathematically challenging case of relative risk aversion larger than one. For this, we argue that the value function is the unique viscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation. The homogeneity of the value function is then used to reduce the HJB equation by one dimension, which allows us to prove that the value function is even a classical solution thereof. Using this, an optimal strategy is derived and its asymptotic behavior in the large wealth regime is discussed.