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Register a new userStability of the Weak Martingale Optimal Transport Problem
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While many questions in (robust) finance can be posed in the martingale optimal transport (MOT) framework, others require to consider also non-linear cost functionals. Following the terminology of Gozlan, Roberto, Samson and Tetali this corresponds to weak martingale optimal transport (WMOT). In this article we establish stability of WMOT which is important since financial data can give only imprecise information on the underlying marginals. As application, we deduce the stability of the superreplication bound for VIX futures as well as the stability of stretched Brownian motion and we derive a monotonicity principle for WMOT.
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In this paper, we obtain stability results for martingale representations in a very general framework. More specifically, we consider a sequence of martingales each adapted to its own filtration, and a sequence of random variables measurable with respect to those filtrations. We assume that the terminal values of the martingales and the associated filtrations converge in the extended sense, and that the limiting martingale is quasi--left--continuous and admits the predictable representation property. Then, we prove that each component in the martingale representation of the sequence converges to the corresponding component of the martingale representation of the limiting random variable relative to the limiting filtration, under the Skorokhod topology. This extends in several directions earlier contributions in the literature, and has applications to stability results for backward SDEs with jumps and to discretisation schemes for stochastic systems.
We study the problem of bounding path-dependent expectations (within any finite time horizon $d$) over the class of discrete-time martingales whose marginal distributions lie within a prescribed tolerance of a given collection of benchmark marginal distributions. This problem is a relaxation of the martingale optimal transport (MOT) problem and is motivated by applications to super-hedging in financial markets. We show that the empirical version of our relaxed MOT problem can be approximated within $Oleft( n^{-1/2}right)$ error where $n$ is the number of samples of each of the individual marginal distributions (generated independently) and using a suitably constructed finite-dimensional linear programming problem.
The classical duality theory of Kantorovich and Kellerer for the classical optimal transport is generalized to an abstract framework and a characterization of the dual elements is provided. This abstract generalization is set in a Banach lattice $cal{X}$ with a order unit. The primal problem is given as the supremum over a convex subset of the positive unit sphere of the topological dual of $cal{X}$ and the dual problem is defined on the bi-dual of $cal{X}$. These results are then applied to several extensions of the classical optimal transport.
We propose two deep neural network-based methods for solving semi-martingale optimal transport problems. The first method is based on a relaxation/penalization of the terminal constraint, and is solved using deep neural networks. The second method is based on the dual formulation of the problem, which we express as a saddle point problem, and is solved using adversarial networks. Both methods are mesh-free and therefore mitigate the curse of dimensionality. We test the performance and accuracy of our methods on several examples up to dimension 10. We also apply the first algorithm to a portfolio optimization problem where the goal is, given an initial wealth distribution, to find an investment strategy leading to a prescribed terminal wealth distribution.
In this paper, we consider Strassens version of optimal transport (OT) problem. That is, we minimize the excess-cost probability (i.e., the probability that the cost is larger than a given value) over all couplings of two given distributions. We derive large deviation, moderate deviation, and central limit theorems for this problem. Our proof is based on Strassens dual formulation of the OT problem, Sanovs theorem on the large deviation principle (LDP) of empirical measures, as well as the moderate deviation principle (MDP) and central limit theorems (CLT) of empirical measures. In order to apply the LDP, MDP, and CLT to Strassens OT problem, two nested optimal transport formulas for Strassens OT problem are derived. Based on these nested formulas and using a splitting technique, we carefully design asymptotically optimal solutions to Strassens OT problem and its dual formulation.
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