Before revision:  To overcome its path-dependent nature, the consumption running maximum process is chosen as an auxiliary state process that renders the value function two dimensional depending on the wealth variable x and the reference variable h. The associated Hamilton-Jacobi-Bellman (HJB) equation is expressed in the piecewise manner across different regions to take into account  constraints.
After revision:  To overcome its path-dependent nature, the consumption running maximum process is chosen as an auxiliary state process that renders the value function two dimensional . The Hamilton-Jacobi-Bellman (HJB) equation is expressed in the piecewise manner across different regions to take into account constraints.