Time-consistent consumption-portfolio control problems with regime-switching-modulated habit formation: an essentially cooperative approach

This paper is devoted to consumption-portfolio control problems with regime-switching-modulated habit formation. The formation of a habit depends on the current regime, which leads to time inconsistency in optimal control. General utility functions are used to evaluate the preference for the consumption net of the habit. To derive an analytical solution, we consider an enlarged financial market with Markov jump assets. Then, the time-inconsistent problem for pre-commitment controls is reduced to solving a regime-switching Hamilton–Jacobi–Bellman (HJB) equation. In seeking a time-consistent control, we investigate an alternate problem by means of multi-person sequential games with an essentially cooperative approach. The analytical expression of the time-consistent control is derived via a straightforward application of the pre-commitment control results. Moreover, the limit case in which the mesh size of time approaches zero is studied.