Stochastic Optimal Transportation

Stochastic Optimal Transportation
Author :
Publisher : Springer Nature
Total Pages : 129
Release :
ISBN-10 : 9789811617546
ISBN-13 : 9811617546
Rating : 4/5 (546 Downloads)

Book Synopsis Stochastic Optimal Transportation by : Toshio Mikami

Download or read book Stochastic Optimal Transportation written by Toshio Mikami and published by Springer Nature. This book was released on 2021-06-15 with total page 129 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 Optimal Transportation Related Books

Stochastic Optimal Transportation
Language: en
Pages: 129
Authors: Toshio Mikami
Categories: Mathematics
Type: BOOK - Published: 2021-06-15 - Publisher: Springer Nature

DOWNLOAD EBOOK

In this book, the optimal transportation problem (OT) is described as a variational problem for absolutely continuous stochastic processes with fixed initial an
Stochastic Optimal Transportation: Stochastic optimal transportation problem
Language: en
Pages: 0
Authors: Toshio Mikami
Categories:
Type: BOOK - Published: 2021 - Publisher:

DOWNLOAD EBOOK

In this book, the optimal transportation problem (OT) is described as a variational problem for absolutely continuous stochastic processes with fixed initial an
Optimal Transportation Problem by Stochastic Optimal Control
Language: en
Pages: 17
Authors:
Categories:
Type: BOOK - Published: 2005 - Publisher:

DOWNLOAD EBOOK

Optimal Transportation Problem by Stochastic Optimal Control
Language: en
Pages: 17
Authors: Toshio Mikami
Categories:
Type: BOOK - Published: 2005 - Publisher:

DOWNLOAD EBOOK

Optimal Transportation and Applications
Language: en
Pages: 176
Authors: Luigi Ambrosio
Categories: Mathematics
Type: BOOK - Published: 2003-01-01 - Publisher: Springer

DOWNLOAD EBOOK

Leading researchers in the field of Optimal Transportation, with different views and perspectives, contribute to this Summer School volume: Monge-Ampère and Mo