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This paper considers a consumption and investment decision problem for a single agent. Wealth is divided between a riskless asset and a risky asset with logarithmic Brownian motion price fluctuations. Short-selling is not allowed, but borrowing is allowed at rate exceeding the rate of return on the riskless asset. An explicit solution of the dynamic programming differential equation for the maximum total discounted expected utility function U is available only in the HARA case. However, using viscosity solution methods the asymptotic behavior of the value function v(x) is found for small wealth x and for large wealth x.
Fleming et al. (Fri,) studied this question.