Optimal Portfolio Selection with Regime-Switching Hamilton-Jacobi-Bellman (HJB) Equation and Maximum Value-at-Risk (MVaR) Constraint

The forming of portfolio is necessary to determine the decision of the best investment so as to investors can identify the securities and determine the allocation of asset to obtain an optimal portfolio. The problem in forming optimal portfolio is the determination of the proportion which is allocated at investment assets in order to maximize expected return with certain risks. The model contains regime-switching market models, which states are interpreted as the states of economy. The risk measuring instrument used is VaR and MVaR is defined as the maximum value of the VaRs in all economy states. The optimal proportion formula is sought by using the stochastic optimal control theory with the aim of maximizing the discounted utility of consumption over a finite time horizon. We use regime-switching Hamilton-Jacobi-Bellman (HJB) equation and then derive a system of coupled HJB equation corresponding to the economy states. Lagrange multiplier method is used to solve the optimization problem with the constraint. We apply Kuhn-Tucker conditions due to MVaR is an inequality function, so that we derive the optimal investment and the optimal consumption. Finally, numerical examples are investigated, and the effect of parameter on the optimal investment and on the optimal consumption are studied.