Author(s)

Lusungu Julius Mbigili¹, Sure Mataramvura², Wilson M. Charles³

¹Department of Mathematics and Statistics Studies, Mzumbe University, Morogoro, Tanzania.
²Actuarial Science Division, University of Cape Town, Cape Town, South Africa.
³Department of Mathematics, University of Dar es Salaam, Dar es Salaam, Tanzania.

This work presented and solved the problem of portfolio optimization within the context of continuous-time stochastic model of financial variables. It has considered an investment problem of two assets, namely, risk-free assets and risky assets. 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 optimal portfolio problem were 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. Two utility functions which are members of hyperbolic absolute risk aversion (HARA) family have been employed, and these are power utility and exponential utility. In both cases, the optimal control has explicit form and is wealth dependent Linearization of the logarithmic term in the portfolio problem was necessary for simplification of the problem.

Portfolio Optimization, Geometric Mean Reversion, Utility Functions and Optimal Policy

Mbigili, L. , Mataramvura, S. and Charles, W. (2020) Optimal Portfolio Management When Stocks Are Driven by Mean Reverting Processes. Journal of Mathematical Finance, 10, 10-26. doi: 10.4236/jmf.2020.101002.