Before revision:  We establish existence_{+}^{0}(\Omega\times[0,T],O,}\mathbb{P} and uniqueness of the optimal solution  using convex duality by defining the primal value function as depending on two variables, i.e., the initial wealth and the initial standard of living.
After revision:  We establish existence_{+}^{0}(\Omega\times[0,T],O,}\mathbb{P} and uniqueness of the optimal solution using convex duality by defining the primal value function is defined on two variables, i.e., the initial wealth and the initial standard of living.