Let \Omega be one of%DIFDELCMD < \X%%% ^{N+1 0,1%DIFDELCMD < ]%%% ,D 0,1%DIFDELCMD < ]%%% : product of Polish spaces, space of continuous functions from 0,1%DIFDELCMD < ] %%% to \mathbb{R 0,1%DIFDELCMD < ] %%% to \mathbb{R^d, respectively. We first consider the existence of a probability measure P on \Omega such that P has the given marginals \alpha and \beta and its disintegration P_x must be in some fixed \Gamma(x) \subset }%DIFDELCMD < \kP%%% (\Omega), where%DIFDELCMD < \kP%%% (\Omega) is the set of probability measures on \Omega. The main application we have in mind is the martingale optimal transport problem when the martingales are assumed to have bounded volatility/ quadratic variation. We show that such probability measure exists if and only if the \alpha average of the so-called G-expectation of bounded continuous functions with respect to the measures in \Gamma is less than their \beta average. As a byproduct, we get a necessary and sufficient condition for the Skorokhod embedding for bounded stopping times. Second, we consider the optimal transport problem with constraints and obtain the Kantorovich duality. A corollary of this result is a monotonicity principle which gives us a geometric way of identifying the optimizer.
%DIFDELCMD < \X%%% %DIFDELCMD < ]%%% %DIFDELCMD < ]%%% %DIFDELCMD < ] %%% %DIFDELCMD < ] %%% ^d, respectively. We first consider the existence of a probability measure P on \Omega such that P has the given marginals \alpha and \beta and its disintegration P_x must be in some fixed \Gamma(x) \subset }%DIFDELCMD < \kP%%% %DIFDELCMD < \kP%%% This paper focuses on martingale optimal transport problems when the martingales are assumed to have bounded quadratic variation. First, we give a result that characterizes the existence of a probability measure satisfying some convex transport constraints in addition to having given initial and terminal marginals. Several applications are provided: martingale measures with volatility uncertainty, optimal transport with capacity constraints, and Skorokhod embedding with bounded times. Next, we extend this result to multi-marginal constraints. Finally, we consider an optimal transport problem with constraints and obtain its Kantorovich duality. A corollary of this result is a monotonicity principle which gives a geometric way of identifying the optimizer.
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We show that such probability measure exists if and only if the \alpha average of the so-called G-expectation of bounded continuous functions with respect to the measures in \Gamma is less than their \beta average. As a byproduct, we get a necessary and sufficient condition for the Skorokhod embedding for bounded stopping times. Second, we consider the
First, we give a result that characterizes the existence of a probability measure satisfying some convex transport constraints in addition to having given initial and terminal marginals. Several applications are provided: martingale measures with volatility uncertainty, optimal transport with capacity constraints, and Skorokhod embedding with bounded times. Next, we extend this result to multi-marginal constraints. Finally, we consider an
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