Before revision:  Using the one-period duality result in bartl2016exponential, together with measurable selection arguments and minimax theorem, we are able to prove all together the existence of the optimal strategy , convex duality theorem as well as the auxiliary dynamic programming principle in our context with transaction costs.
After revision:  By suggesting a different dynamic programming argument than in bartl2016exponential, together with measurable selection arguments and minimax theorem, we are able to prove all together the existence of the optimal strategy , convex duality theorem as well as the auxiliary dynamic programming principle in our context with transaction costs.