Optimal Consumption and Investment Strategies with Liquidity Risk and Lifetime Uncertainty for Markov Regime-Switching Jump Diffusion Models

In this paper, we consider the optimal consumption and investment strategies for households throughout their lifetime. Risks such as the illiquidity of assets, abrupt changes of market states, and lifetime uncer- tainty are considered. Taking the effects of heritage into account, investors are willing to limit their cur- rent consumption in exchange for greater wealth at their death, because they can take advantage of the higher expected returns of illiquid assets. Further, we model the liquidity risks in an illiquid market state by introducing frozen periods with uncertain lengths, during which investors cannot continuously rebal- ance their portfolios between different types of assets. In liquid market, investors can continuously remix their investment portfolios. In addition, a Markov regime-switching process is introduced to describe the changes in the market’s states. Jumps, classified as either moderate or severe, are jointly investigated with liquidity risks. Explicit forms of the optimal consumption and investment strategies are developed using the dynamic programming principle. Markov chain approximation methods are adopted to obtain the value function. Numerical examples demonstrate that the liquidity of assets and market states have significant effects on optimal consumption and investment strategies in various scenarios.