Optimal Investment-Consumption Strategies in Diffusion Models

Abstract

In complete markets with diffusion models, the expected utility maximization problem has been studied by many authors (see a review in Karatzas 1989). Karatzas et al. (1991) studied this problem in incomplete markets. They considered a market composed of a bond and d stocks, the latter being driven by an m-dimensional Brownian motion. They used some virtual stocks to expand the original market into a complete market. Under certain additional conditions, they proved that one can wisely choose virtual stocks, such that in the resulting optimal portfolio for the solution to utility maximization problem in the completed market, virtual stocks are superfluous. Thus, this solution is also the optimal one in the original incomplete market.