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.