Before revision:  (iii) \lambda h\leq c^*(x,h)<h when (h)\leq x<x_2(h); (iv) c^*(x,h)=h but the running maximum process remains flat when x_2(h)\leq x<x_3(h); (v) c^*(x,h)=h and the running maximum process increases when x=x_3(h). Similar conclusions can be made in a simpler fashion for two extreme cases \lambda=0 and \lambda=1. Numerical examples are also presented to illustrate some theoretical results and financial insights .
After revision:  (iii) \lambda h\leq c^*(x,h)<h when strategies can be obtained in the closed form in each region. The complete proof of the verification theorem is provided and numerical examples are presented to illustrate some theoretical results and financial insights .