Optimal investment-consumption strategy with liability and regime switching model under Value-at-Risk constraint

This paper considers the optimal investment-consumption problem with liability subject to a maximum Value-at-Risk (denoted by MVaR) constraint. The model also contains regime-switching market modes, whose states are interpreted as the states of the economy. In each state of economy state, we constrain a VaR value for the portfolio in a short time duration, and MVaR is defined as the maximum value of the VaRs in all economy states. We suppose that both the price dynamics of the risky asset and the liability value process are governed by a Markov-modulated geometric Brownian motion. With the objective of maximizing the discounted utility of consumption, we obtain a system of HJB equations corresponding to the economy states by using the dynamic programming principle. Then, by adopting the techniques of Chen et al. (2010), we get explicit expressions of value functions. Moreover, with the help of Lagrange multiplier method, we derive the optimal investment and the optimal consumption. Finally, a numerical example is investigated, and the effects of many parameters on the optimal investment, on the optimal consumption and on VaR value are studied. Furthermore, we also explore how VaR value affects the optimal investment and the optimal consumption.