A consumption-investment model with state-dependent lower bound constraint on consumption

This paper studies a life-time consumption-investment problem under the Black-Scholes framework, where consumption rate is subject to lower bound constraint that linearly depends on investor's wealth. It stochastic control with state-dependent which standard theory cannot be directly applied. We overcome this by transforming it into an equivalent in state-independent so can give explicit optimal strategy when homogeneous. When non-homogeneous, shown value function third-order continuously differentiable differential equation approach, and feedback form provided. According our findings, if investor concerned long-term more than short-term consumption, then she should always consume as few possible; otherwise, optimally her wealth above threshold, possible below threshold.