Optimal Consumption, Investment, and Insurance in Multi-State Path-Dependent Models

We study how past consumption and health history affect optimal consumption, investment, and insurance in a multi-state life-insurance model. Two one-dimensional indices encode these distinct forms of history. The Habit-index is controlled by consumption and enters preferences, whereas the Health-index is uncontrolled and may also determine biometric transition intensities; state duration is obtained as a special case. Under power utility, both formulations admit explicit feedback controls and Feynman–Kac representations but lead respectively to systems of ordinary and partial differential equations. We give a verification theorem for the augmented-state control problem. A stylised three-state disability example isolates the mechanism through which additive habit formation can generate a consumption hump.