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ABSTRACT

We considered a simple optimization problem for an investor whose actions cannot affect the market prices and have no other profit than the returns on the financial investments. Under general conditions on the nature of the market model, the optimal consumption and the optimal investment were derived from our wealth process. Risky asset price () obeys a geometric Brownian motion. The method used is dynamic programming principle which is used to derive the Hamilton-Jacobi-Bellman equation (HJB) which is a second order non-linear differential equation. In addition we analyzed the optimal consumption of an investor for 10years and our result shows that the optimal consumption of the investor is to consume a fraction of his wealth which also coincides with the wealth at terminal. We also focused on analyzing the optimal bond/stock mix for a single stock. Our result shows that for a logarithmic investor, investing large part of their wealth on stock tends to yield higher return. Illustrative example is given.

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