A Dissertation Submitted in Partial Ful lment of the Requirements for the Degree of Master of Science(Mathematical Modelling) of the University of Dar es Salaam

In this work, we present and solve the problem of portfolio optimization within the context of continuous-time stochastic model of nancial variables. We con- sider an investment problem where an investor has two assets, namely, risk-free assets(eg bonds) and risky assets(eg stocks) to invest on and tries to maximize the expected utility of the wealth at some future time . The evolution of the risk-free asset is described deterministically while the dynamics of the risky asset is described by the geometric mean reversion(GMR) model.
The controlled wealth stochastic di erential equation(SDE) and the portfolio problem are formulated. The portfolio optimization problem is then successfully formulated and solved with the help of the theory of stochastic control technique where the dynamic programming principle(DPP) and the HJB theory were used. We obtained very interesting results which are the solution of the non-linear sec- ond order partial di erential equation and the optimal policy which is the optimal control strategy for the investment process.
So far we have considered utility functions which are members of hyperbolic ab- solute risk aversion(HARA) family, called power and exponential utility. In both cases, the optimal control(investment strategy) has explicit form and is wealth dependant, in the sense that, as the investor becomes more rich, the less he in- vests on the risky assets. Linearization of the logarithmic term in the portfolio problem was necessary to be undertaken for making the work of obtaining the explicit form of the optimal control much simple than it was expected.