Before revision:  We also provide market independent sufficient conditions on both 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.
After revision:  We also provide market independent sufficient conditions on both stochastic discounting processes of the habit formation process and on the utility function for the well-posedness of our original optimization problem. Under the same assumptions, we can carefully modify the classical proofs in the approach of convex duality analysis when the auxiliary dual process is not necessarily integrable.