We develop computational methods for solving the martingale optimal transport (MOT) problem -a version of the classical optimal transport with an additional martingale constraint on the transport's dynamics. We prove that a general, multi-step multi-dimensional, MOT problem can be approximated through a sequence of linear programming (LP) problems which result from a discretization of the marginal distributions combined with an appropriate relaxation of the martingale condition. We further furnish two generic approaches for discretizing probability distributions, suitable respectively for the cases when we can compute integrals against these distributions or when we can sample from them. These render our main result applicable and lead to an implementable numerical scheme for solving MOT problems. Finally, specializing to the one-step model on real line, we provide an estimate of the convergence rate which, to the best of our knowledge, is the first of its kind in the literature. for insightful discussions and comments. † The second author also gratefully acknowledges support from St John's College, Oxford. MSC 2010 subject classifications: Primary 49M25, 60H99; secondary 90C08. 1 imsart-aap ver.
The Skorokhod embedding problem aims to represent a given probability measure on the real line as the distribution of Brownian motion stopped at a chosen stopping time. In this paper, we consider an extension of the weak formulation of the optimal Skorokhod embedding problem in Beiglböck, Cox & Huesmann [1] to the case of finitely-many marginal constraints 1 . Using the classical convex duality approach together with the optimal stopping theory, we establish some duality results under more general conditions than [1]. We also relate these results to the problem of martingale optimal transport under multiple marginal constraints.
The martingale optimal transport aims to optimally transfer a probability measure to another along the class of martingales. This problem is mainly motivated by the robust superhedging of exotic derivatives in financial mathematics, which turns out to be the corresponding Kantorovich dual. In this paper we consider the continuoustime martingale transport on the Skorokhod space of càdlàg paths. Similar to the classical setting of optimal transport, we introduce different dual problems and establish the corresponding dualities by a crucial use of the S−topology and the dynamic programming principle 1 . 1 During the preparation of the final version of this paper, we knew from Mete Soner about the latest version of Dolinsky & Soner [24] which includes the finitely-many marginal constraints setting.where (X, Y ) denotes the canonical process on R d × R d , i.e. X(x, y) = x and Y (x, y) = y for all (x, y) ∈ R d × R d . Then the optimal transport problem consists in optimizing the expectation of some measurable function ξ : R d × R d → R among all probability measures in P(µ, ν). Various related issues are studied, e.g. the general duality theory and optimality results, we refer to Rachev & Rüschendorf [53] and Villani [55] for a comprehensive account of the literature.Recently, a martingale optimal transport problem was introduced in Beiglböck, Henry-Labordère & Penkner [5] in discrete-time (see Galichon, for the continuous-time case), where a maximization problem is considered over a subset M(µ, ν) := P ∈ P(µ, ν) : E P [Y |X] = X, P-a.s. :Each element of M(µ, ν) is called a transport plan. Similarly to the classical setting, the corresponding dual problem is defined byThe last dual formulation has the interpretation of minimal robust superhedging cost of derivative security defined by the payoff ξ by trading the underlying security and any possible Vanilla option. When d = 1, as observed by Breeden & Litzenberger [11], the marginal distributions of the underlying asset are recovered by the market prices of calls for all strikes, and any Vanilla option has a non-ambiguous price as the integral of its payoff function with respect to the marginal. Therefore, the inequality (1.2) represents a super-replication of ξ, which consist of the trading of the underlying and Vanilla options at different maturities. Since there is no specific model imposed on the process (X, Y ), the dual problem may be interpreted as the robust superhedging cost, i.e. the minimum cost to construct super-replications. Similar to the classical setting, the duality P(µ, ν) = D(µ, ν) holds under quite general conditions. The present paper considers the continuous-time martingale optimal transport problem. Let X := ω = (ω t ) 0≤t≤1 : ω t ∈ R d for all t ∈ [0, 1] , where X is either the space of continuous functions or the Skorokhod space of càdlàg functions. Denote by X = (X t ) 0≤t≤1 the canonical process and by M the set of all martingale measures P, i.e. X is a martingale under P. For a given family of probability measures µ = (µ t ) t∈T , where T...
This is a continuation of our accompanying paper [18]. We provide an alternative proof of the monotonicity principle for the optimal Skorokhod embedding problem established in Beiglböck, Cox & Huesmann [2]. Our proof is based on the adaptation of the Monge-Kantorovich duality in our context, a delicate application of the optional cross-section theorem, and a clever conditioning argument introduced in [2].
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