Optimal portfolio management when stocks are driven by mean-reverting process

In this work, we present and solve the problem of portfolio optimization within the context of continuous-time stochastic model of financial 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 r. 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 differential 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 results which are the solution of the non-linear second order partial differential equation and the optimal policy which is the optimal control strategy for the investment process. We considered utility functions which are members of hyperbolic absolute risk aversion(HARA) family, called power and exponential utility. In both cases, the optimal control (investment strategy) has explicit form and is wealth dependent, in the sense that, as the investor becomes more rich, the less he invests on the risky assets. Linearization of the logarithmic term in the portfolio problem was necessary for making the work of obtaining the explicit form of the optimal con¬trol much simpler than it was expected.