### Solutions to some portfolio optimization problems with stochastic income and consumption

#### Abstract

We solve an optimal portfolio choice problem for an investor with either power or log utility over terminal wealth in close form, facing imperfectly hedgeable stochastic income. The returns on the income and the stock are imperfectly correlated, therefore the market is incomplete. We describe how an investor accommodates or adjusts the Merton portfolio of the stock and risk-free asset through an interpolating hedging demand, in reaction to the stochastic income. The solutions to the investor thrilling problem of seeking the optimal portfolio are formulated and worked out using the stochastic control theory. The Bellman principle of dynamic optimality is utilized through the Hamilton-Jacobi-Bellman (HJB) partial differential equation. We apply the results to some unconstrained portfolio optimization problem with power and log utility functions which lead to four propositions as the main results. All the two models discussed shows that, there is an inverse relation between the risk and the value of Merton’s investment strategy.

**Published:**2015-10-28

**How to Cite this Article:**Obonye Doctor, Elias Rabson Offen, Solutions to some portfolio optimization problems with stochastic income and consumption, Math. Finance Lett., 2015 (2015), Article ID 9 Copyright © 2015 Obonye Doctor, Elias Rabson Offen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Mathematical Finance Letters

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