Equilibrium Strategies for Time-Inconsistent Stochastic Optimal Control of Asset Allocation

Sammanfattning: We have examinined the problem of constructing efficient strategies for continuous-time dynamic asset allocation. In order to obtain efficient investment strategies; a stochastic optimal control approach was applied to find optimal transaction control. Two mathematical problems are formulized and studied: Model I; a dynamic programming approach that maximizes an isoelastic functional with respect to given underlying portfolio dynamics and Model II; a more sophisticated approach where a time-inconsistent state dependent mean-variance functional is considered. In contrast to the optimal controls for Model I, which are obtained by solving the Hamilton-Jacobi-Bellman (HJB) partial differential equation; the efficient strategies for Model II are constructed by attaining subgame perfect Nash equilibrium controls that satisfy the extended HJB equation, introduced by Björk et al. in [1]. Furthermore; comprehensive execution algorithms where designed with help from the generated results and several simulations are performed. The results reveal that optimality is obtained for Model I by holding a fix portfolio balance throughout the whole investment period and Model II suggests a continuous liquidation of the risky holdings as time evolves. A clear advantage of using Model II is concluded as it is far more efficient and actually takes time-inconsistency into consideration.