Before revision:  Introducing the set of auxiliary state processes and the modified dual space, we embed our original problem into an abstract time-separable utility maximization problem with a shadow random endowment on the product space . 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:  Introducing the set of auxiliary state processes and the modified dual space, we embed our original problem into an abstract time-separable utility maximization problem with a shadow random endowment on the product space \mathbb{L_{+}^{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.