Stochastic Optimal TransportationStochastic Optimal Transportation

In this book, the optimal transportation problem (OT) is described as a variational problem for absolutely continuous stochastic processes with fixed initial and terminal distributions.

Author: Toshio Mikami

Publisher: Springer Nature

ISBN: 9789811617546


Page: 121

View: 750

In this book, the optimal transportation problem (OT) is described as a variational problem for absolutely continuous stochastic processes with fixed initial and terminal distributions. Also described is Schrödinger’s problem, which is originally a variational problem for one-step random walks with fixed initial and terminal distributions. The stochastic optimal transportation problem (SOT) is then introduced as a generalization of the OT, i.e., as a variational problem for semimartingales with fixed initial and terminal distributions. An interpretation of the SOT is also stated as a generalization of Schrödinger’s problem. After the brief introduction above, the fundamental results on the SOT are described: duality theorem, a sufficient condition for the problem to be finite, forward–backward stochastic differential equations (SDE) for the minimizer, and so on. The recent development of the superposition principle plays a crucial role in the SOT. A systematic method is introduced to consider two problems: one with fixed initial and terminal distributions and one with fixed marginal distributions for all times. By the zero-noise limit of the SOT, the probabilistic proofs to Monge’s problem with a quadratic cost and the duality theorem for the OT are described. Also described are the Lipschitz continuity and the semiconcavity of Schrödinger’s problem in marginal distributions and random variables with given marginals, respectively. As well, there is an explanation of the regularity result for the solution to Schrödinger’s functional equation when the space of Borel probability measures is endowed with a strong or a weak topology, and it is shown that Schrödinger’s problem can be considered a class of mean field games. The construction of stochastic processes with given marginals, called the marginal problem for stochastic processes, is discussed as an application of the SOT and the OT.

Stochastic Analysis Filtering and Stochastic OptimizationStochastic Analysis Filtering and Stochastic Optimization

Singular Perturbation Methods in Stochastic Differential Equations of Mathematical Physics. SIAM Rev., 22(2):119–155, 1980. 44. X. Tan and N. Touzi. Optimal transportation under controlled stochastic dynamics.

Author: George Yin

Publisher: Springer Nature

ISBN: 9783030985196


Page: 466

View: 261

This volume is a collection of research works to honor the late Professor Mark H.A. Davis, whose pioneering work in the areas of Stochastic Processes, Filtering, and Stochastic Optimization spans more than five decades. Invited authors include his dissertation advisor, past collaborators, colleagues, mentees, and graduate students of Professor Davis, as well as scholars who have worked in the above areas. Their contributions may expand upon topics in piecewise deterministic processes, pathwise stochastic calculus, martingale methods in stochastic optimization, filtering, mean-field games, time-inconsistency, as well as impulse, singular, risk-sensitive and robust stochastic control.

Multistage Stochastic OptimizationMultistage Stochastic Optimization

Given two probability spaces . ;F;P/ and Q; QF; QP and a transportation cost function c, the optimal transportation cost is inf “ Q c .!; Q!/ .d!;d Q!/; (2.8) where the infimum is taken over all (bivariate) probability measures on Q ...

Author: Georg Ch. Pflug

Publisher: Springer

ISBN: 9783319088433


Page: 301

View: 453

Multistage stochastic optimization problems appear in many ways in finance, insurance, energy production and trading, logistics and transportation, among other areas. They describe decision situations under uncertainty and with a longer planning horizon. This book contains a comprehensive treatment of today’s state of the art in multistage stochastic optimization. It covers the mathematical backgrounds of approximation theory as well as numerous practical algorithms and examples for the generation and handling of scenario trees. A special emphasis is put on estimation and bounding of the modeling error using novel distance concepts, on time consistency and the role of model ambiguity in the decision process. An extensive treatment of examples from electricity production, asset liability management and inventory control concludes the book.

Convex and Stochastic OptimizationConvex and Stochastic Optimization

On transportation theory, see the monographs by Villani [122] and Santambrogio [108]. The link (9.80) between a ... Augmented Lagrangians are a useful numerical tool for solving optimal transport problems, see Benamou and Carlier [17].

Author: J. Frédéric Bonnans

Publisher: Springer

ISBN: 9783030149772


Page: 311

View: 705

This textbook provides an introduction to convex duality for optimization problems in Banach spaces, integration theory, and their application to stochastic programming problems in a static or dynamic setting. It introduces and analyses the main algorithms for stochastic programs, while the theoretical aspects are carefully dealt with. The reader is shown how these tools can be applied to various fields, including approximation theory, semidefinite and second-order cone programming and linear decision rules. This textbook is recommended for students, engineers and researchers who are willing to take a rigorous approach to the mathematics involved in the application of duality theory to optimization with uncertainty.

Optimal Transportation NetworksOptimal Transportation Networks

1851: O. Catoni, Statistical Learning Theory and Stochastic Optimization (2004) Vol. 1852: A.S. Kechris, B.D. Miller, Topics in Orbit Equivalence (2004) Vol. 1853: Ch. Favre, M. Jonsson, The Valuative Tree (2004) Vol.

Author: Marc Bernot

Publisher: Springer Science & Business Media

ISBN: 9783540693147


Page: 200

View: 702

The transportation problem can be formalized as the problem of finding the optimal way to transport a given measure into another with the same mass. In contrast to the Monge-Kantorovitch problem, recent approaches model the branched structure of such supply networks as minima of an energy functional whose essential feature is to favour wide roads. Such a branched structure is observable in ground transportation networks, in draining and irrigation systems, in electrical power supply systems and in natural counterparts such as blood vessels or the branches of trees. These lectures provide mathematical proof of several existence, structure and regularity properties empirically observed in transportation networks. The link with previous discrete physical models of irrigation and erosion models in geomorphology and with discrete telecommunication and transportation models is discussed. It will be mathematically proven that the majority fit in the simple model sketched in this volume.

Stochastic OptimizationStochastic Optimization

Seeing the Optimal for the Uncertain Ioannis Dritsas. Falls, L.C. & Tighe, S. (2003). Improving LCCA through the Development of Cost Models Using the Alberta Roadway Rehabilitation and Maintenance Analysis. Transportation Research Board ...

Author: Ioannis Dritsas

Publisher: BoD – Books on Demand

ISBN: 9789533078298


Page: 490

View: 892

Stochastic Optimization Algorithms have become essential tools in solving a wide range of difficult and critical optimization problems. Such methods are able to find the optimum solution of a problem with uncertain elements or to algorithmically incorporate uncertainty to solve a deterministic problem. They even succeed in fighting uncertainty with uncertainty. This book discusses theoretical aspects of many such algorithms and covers their application in various scientific fields.

Optimal TransportOptimal Transport

reveals the conditional probability Px of the transportation from x e X to B C Y. Thus , we can interpret Kantorovich's transport plan as a stochastic transport . In contrast , deterministic transport T via Monge's paradigm is the ...

Author: Gershon Wolansky

Publisher: Walter de Gruyter GmbH & Co KG

ISBN: 9783110635485


Page: 224

View: 914

This book focuses on problems at the interplay between the theory of partitions and optimal transport with a view toward applications. Topics covered include problems related to stable marriages and stable partitions, multipartitions, optimal transport for measures and optimal partitions, and finally cooperative and noncooperative partitions. All concepts presented are illustrated by examples from game theory, economics, and learning.

Stochastic Optimization Methods in Finance and EnergyStochastic Optimization Methods in Finance and Energy

(PI, Q1) ||(1 + w, K.) + X) r, w, TI(1 + w; Kj) (1533) S= 1 S= 1 j=1 Proof Let It" denote the optimal joint measure (optimal transportation plan) for d (P', Q") and let T-1||u, v] denote the optimal joint measure for d; (P)+1. u], ...

Author: Marida Bertocchi

Publisher: Springer Science & Business Media

ISBN: 1441995862


Page: 476

View: 943

This volume presents a collection of contributions dedicated to applied problems in the financial and energy sectors that have been formulated and solved in a stochastic optimization framework. The invited authors represent a group of scientists and practitioners, who cooperated in recent years to facilitate the growing penetration of stochastic programming techniques in real-world applications, inducing a significant advance over a large spectrum of complex decision problems. After the recent widespread liberalization of the energy sector in Europe and the unprecedented growth of energy prices in international commodity markets, we have witnessed a significant convergence of strategic decision problems in the energy and financial sectors. This has often resulted in common open issues and has induced a remarkable effort by the industrial and scientific communities to facilitate the adoption of advanced analytical and decision tools. The main concerns of the financial community over the last decade have suddenly penetrated the energy sector inducing a remarkable scientific and practical effort to address previously unforeseeable management problems. Stochastic Optimization Methods in Finance and Energy: New Financial Products and Energy Markets Strategies aims to include in a unified framework for the first time an extensive set of contributions related to real-world applied problems in finance and energy, leading to a common methodological approach and in many cases having similar underlying economic and financial implications. Part 1 of the book presents 6 chapters related to financial applications; Part 2 presents 7 chapters on energy applications; and Part 3 presents 5 chapters devoted to specific theoretical and computational issues.