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:  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 and the convex duality theorem as well as the auxiliary dynamic programming principle in our context with transaction costs.