Stochastic Optimal Transportation
Stochastic Control with Fixed Marginals
Authors: Mikami, Toshio
Free Preview Shows the SOT problem to be partly the generalization of the OT problem and partly Schrödinger's problem
 Explains fundamental results of the stochastic optimal transportation problem, including duality theorem
 Encompasses the zeronoise limit, the Lipschitz continuity, and the semiconcavity of Schrödinger's problem
Buy this book
 About this book

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 onestep 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 zeronoise 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.
 Table of contents (3 chapters)


Introduction
Pages 119

Stochastic Optimal Transportation Problem
Pages 2175

Marginal Problem
Pages 77113

Table of contents (3 chapters)
Recommended for you
Bibliographic Information
 Bibliographic Information

 Book Title
 Stochastic Optimal Transportation
 Book Subtitle
 Stochastic Control with Fixed Marginals
 Authors

 Toshio Mikami
 Series Title
 SpringerBriefs in Mathematics
 Copyright
 2021
 Publisher
 Springer Singapore
 Copyright Holder
 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
 eBook ISBN
 9789811617546
 DOI
 10.1007/9789811617546
 Softcover ISBN
 9789811617539
 Series ISSN
 21918198
 Edition Number
 1
 Number of Pages
 XI, 121
 Number of Illustrations
 15 b/w illustrations
 Topics