Degenerate diffusion processes in portfolio management

The method of variational inequalities is a useful theoretical tool in stochastic control, but there are few problems in which this method leads to an explicit solution. We present such a problem drawn from portfolio management. An agent can distribute his wealth between two investments, one risky and the other risk-free. He can also consume his wealth, and wishes to manage his portfolio and consumption to maximize total, discounted, expected utility of consumption. This is a degenerate diffusion control problem which can be solved completely. The solution satisfies a related variational inequality.