Before revision:  This allows us to prove all together the existence of the optimal strategy,  the auxiliary dynamic programming principle as well as the convex duality theorem in our context with transaction costs.
After revision:  Using the one-period duality result in \mbox{%DIFAUXCMD bartl2016exponential to prove all together the existence of the optimal strategy, the auxiliary dynamic programming principle as well as the convex duality theorem in our context with transaction costs.