Before revision:  We establish existence and uniqueness of the optimal solution using convex duality approach on the product space _{+}^{0}(\Omega\times[0,T],O,}\mathbb{P} by defining the primal value function both on the initial wealth and initial habit .
After revision:  We establish existence and uniqueness of the optimal solution using convex duality on the product space _{+}^{0}(\Omega\times[0,T],O,}\mathbb{P} by defining the primal value function both on the initial wealth and initial habit .