Consumption-Portfolio Optimization with Regime- Switching-Modulated Habit Formation and Jump Diffusion

This paper studies consumption-portfolio optimization problems with habit formation in a regime-switching market. The habit level, which reflects the endogenous impact of past consumption, also involves regime switching and jump diffusion. Because of the presence of general utility functions and path-dependent random parameters, we use the market completion method and introduce some additional jump assets to address the problems. After reducing the problems to solving a stochastic Hamilton-Jacobi-Bellman equation, we derive the optimal control by a joint adoption of envelope theorem and backward stochastic partial differential equation. In general, the optimal portfolio strategy includes the demand of jump assets for hedging against the regime-switching and jump-diffusion risk. In particular, for power/logarithmic utility, we obtain a closed-form solution in the enlarged complete market. For comparison, we also study the power/logarithmic utility case with many specific conditions in the primal incomplete market by restricting the positions of the jump assets to zero. Funding: This research was supported by the National Natural Science Foundation of China [Grants 12401611 and 12571520], Major Program of the Key Research Institute on Humanities and Social Science of China Ministry of Education [Grant 22JJD790091], the 111 Project [Grant B17050], PolyU [Grant 1-CE28], and CTBU [Grant 2355010].