Model-Independent Superhedging with Portfolio Constraints in Continuous Time under Weak S Topology

In this paper we consider the same problem as the one in \cite{F-H-2016} in continuous time. That is a Monge-Kantorovich like problem under some portfolio constraints but the set of transport plans does not have to be the martingale measures. Instead, we consider the set of semi-martingale measures with a marginal distribution at terminal time $T=1$. Since we only assume the distribution of the underlying asset at the terminal time and we do not know any information prior to terminal time, so we exert extra assumptions on the set of semi-martingale measures. The main difficulty comes from proving the lower semi-continuity of the expectation of penalty term $A_1(\Q)$. We use continuous selection theory to extend the penalty term under the weak $S$ topology on Skorohod space, introduced by Kiiski in \cite{Kiiski2008a}, \cite{Kiiski2008b}. The $S$ topology introduced by Jakubowski \cite{jakubowski} is Hausdorff but the regularity of Skorohod space under the $S$ topology is not clear and the continuous selection theory may fail under such topology. The weak $S$ topology has almost all the properties of $S$ topology and the Skorohod space under weak $S$ topology is Hausdorff perfectly normal. So we can then apply the continuous selection theory. Another benefit from the weak $S$ topology is the limit measure $\Q$ preserves the properties of a convergent sequence of semi-martingale measures. Under the weak $S$ topology and the assumptions on payoff function and portfolio constraints we showed upper semi-continuity of the primal problem $P(\mu)=\sup_{\Q\in\cQ(\mu)}\E^\Q[G-A_1(\Q)]$. From there we extend the first duality in \cite{G-T-T-2017} and use the discretization procedure in \cite{Dolinsky2} to proved the super-hedging duality under portfolio constraints in continuous time.