Let \Omega to be one of \X^{N+1},C[0,1],D[0,1]: a product of Polish spaces, space of continuous functions from [0,1] to a subset of R^d, and space of RCLL (right-continuous with left limits) functions from [0,1] to R^d  respectively. We  consider the existence of a probability measure P on \Omega such that P has the given marginals \alpha and \beta and satisfies some other convex transport constraints, which is given by \Gamma\kP\kP . The main application we have in mind is the martingale optimal transport problem with 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  so-called G-expectation of bounded uniformly continuous and bounded 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  .
Let \Omega be one of \X^{N+1},C[0,1],D[0,1]: product of Polish spaces, space of continuous functions from [0,1] to R^d, and space of RCLL (right-continuous with left limits) functions from [0,1] to 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\kP(\Omega), where\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 .
D
to
D
a
D
a subset of
A
,
A
first
R
satisfies some other convex transport constraints, which is given by \Gamma
its disintegration P_x must be in some fixed \Gamma(x) \subset
A
(\Omega), where
A
(\Omega) is the set of probability measures on \Omega
D
with
A
the
R
uniformly continuous and bounded
continuous
A
. 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