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