In probabilistic terms, Optimal Transport (OT) is the theory of extremal couplings between stochastic process. Nevertheless, classical OT is oblivious to the information structure (i.e. filtrations and adaptedness) that make stochastic processes so appealing in many applications in economics, finance, etc. Causal Optimal Transport is what appears when we take information into account in an OT.
Objectives: The thematic period aims to provide an overview of the current state of research in calculus of variations, optimal transportation theory, and geometric measure theory, from both the perspectives of theory and applications.The scope of the conference ranges from rigorous mathematical analysis to modeling, numerical analysis, and scientific computing for real world applications in.Abstract In the rst part of this thesis, we study the structure of solutions to the optimal mar-tingale transport problem, when the marginals lie in higher dimensional Euclidean s.A unifying goal of these authors is to strengthen the usual weak topology in order to adequately capture the temporal structure of stochastic processes. We find that all of these seemingly independent approaches define the same topology in finite discrete time. Moreover, we explain how optimal transport theory can be used obtain a compatible metric that is both tractable and highly relevant.
Section: Research Program Modeling and Analysis. The first layer of methodological tools developed by our team is a set of theoretical continuous models that aim at formalizing the problems studied in the applications. These theoretical findings will also pave the way to efficient numerical solvers that are detailed in Section 3.2. Static Optimal Transport and Generalizations Convexity.
Martingale Optimal Transport in Robust Finance. Abstract Without assuming any probabilistic price dynamics, we consider a frictionless financial market given by the Skorokhod space, on which some financial options are liquidly traded. In this model-free setting we show various pricing-hedging dualities and the analogue of the.
Constructing sampling schemes via coupling: Markov semigroups and optimal transport N. Nuesken, G. A. Pavliotis. In this paper we develop a general framework for constructing and analysing coupled Markov chain Monte Carlo samplers, allowing for both (possibly degenerate) diffusion and piecewise deterministic Markov processes. For many performance criteria of interest, including the asymptotic.
If the martingale is optimal, then each of the components in the decomposition supports a restricted optimal martingale transport for which the dual problem is attained. These decompositions are used to obtain structural results in cases where duality is not attained. On the other hand, they can also be related to higher dimensional Nikodym sets. %On the other hand, they can also lead to.
Optimal Regularity and Structure of the Free Boundary for Minimizers in Cohesive zone models (2020) Y. R. Y. Zhang - A. Figalli (Accepted Paper: Comm. Pure Appl. Math. ) Strong stability for the Wulff inequality with a crystalline norm (2020).
Three key features of the approach are worth pointing out: First, the homogeneous martingale optimal transport problem is as numerically tractable as the martingale optimal transport problem without time-homogeneity, in that the discretized version reduces to a linear program and the dual formulation is well suited for various approaches, see e.g. (11, 14, 16).
Formulation and Duality Motivation Quasi-Sure Formulation Martingale Optimal Transport NizarTouzi EcolePolytechnique,Paris IHP,January10,2017 BonanniversaireYann!
Title: Optimal Transport with a Martingale Constraint: theory, applications and numerics Abstract: Optimal transportation is a very rich and well-established field in mathematics. I consider here its variant where the transport has a direction and an additional martingale, or barycentre preservation, constraint. I will explain how this problem.
I will describe the profile of optimal solutions of the martingale counterpart of the Monge mass transport problem.
Abstract: We solve the martingale optimal transport problem for cost functionals represented by optimal stopping problems. The measure-valued martingale approach developed in ArXiv: 1507.02651 allows us to obtain an equivalent infinite-dimensional controller-stopper problem. We use the stochastic Perron's method and characterize the finite dimensional approximation as a viscosity solution to.
The second part is devoted to the existence and uniqueness of solutions of a general stochastic evolution equation, and the third concerns the qualitative properties of those solutions. Appendices gather together background results from analysis that are otherwise hard to find under one roof. This revised edition includes two brand new chapters surveying recent developments in the area and an.
We introduce a Benamou-Brenier formulation for the continuous-time martingale optimal trans-port problem as a weak length relaxation of its discrete-time counterpart. By the correspondence between classical martingale problems and Fokker-Planck equations, we obtain an equivalent PDE formulation for which basic properties such as existence, duality and geodesic equations.
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