Optimal consumption with stochastic prices in continuous time

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Abstract
In this paper we consider an economic agent (the consumer) having initial wealth x(0), which he is allowed to spend during the time interval [0, T]. There are N consumer goods, the prices of which are described by a system of stochastic differential equations, and the consumer can spend his money as he pleases provided that he does not get into debt. The consumer's problem is to maximize his expected utility over [0, T]. Given a separable utility function we show that the optimal consumption policy is linear in the wealth variable regardless of the structure of the price system, and if we assume that the price processes are geometric Brownian motions we can give the optimal consumption policy explicitly.