Before revision:  Introducing the set of auxiliary state processes and the modified dual space, we embed our original problem into a time-separable utility maximization problem with a shadow random endowment on the product space L _{+^{0}(\Omega\times[0,T],O,}  ],}\mathbb{P} ).
After revision:  Introducing the set of auxiliary state processes and the modified dual space, we embed our original problem into a time-separable utility maximization problem with a shadow random endowment on the product space L _{+^{0}(\Omega\times[0,T],O,} _+^0(\Omega\times ],}\mathbb{P} ).