Before revision:  We also provide market independent sufficient conditions both on stochastic discounting processes of the habit formation process and on the utility function  to modify the convex duality approach when the auxiliary dual process is not necessarily integrable.
After revision:  We also provide market independent sufficient conditions both on stochastic discounting processes of the habit formation process and on the utility function for our original problem to be well posed and to modify the convex duality approach when the auxiliary dual process is not necessarily integrable.