40% off Popular Science books & eBooks—Save on general interest titles now!

SpringerBriefs in Mathematics

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 zero-noise limit, the Lipschitz continuity, and the semiconcavity of Schrödinger's problem
see more benefits

Buy this book

eBook 46,00 €
price for Spain (gross)
  • Due: June 15, 2021
  • ISBN 978-981-16-1754-6
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
Softcover 57,19 €
price for Spain (gross)
  • Due: June 16, 2021
  • ISBN 978-981-16-1753-9
  • Free shipping for individuals worldwide
  • Institutional customers should get in touch with their account manager
  • Covid-19 shipping restrictions
  • The final prices may differ from the prices shown due to specifics of VAT rules
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 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.

Table of contents (3 chapters)

Table of contents (3 chapters)

Buy this book

eBook 46,00 €
price for Spain (gross)
  • Due: June 15, 2021
  • ISBN 978-981-16-1754-6
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
Softcover 57,19 €
price for Spain (gross)
  • Due: June 16, 2021
  • ISBN 978-981-16-1753-9
  • Free shipping for individuals worldwide
  • Institutional customers should get in touch with their account manager
  • Covid-19 shipping restrictions
  • The final prices may differ from the prices shown due to specifics of VAT rules
Loading...

Services for this Book

Recommended for you

Loading...

Bibliographic Information

Bibliographic Information
Book Title
Stochastic Optimal Transportation
Book Subtitle
Stochastic Control with Fixed Marginals
Authors
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
978-981-16-1754-6
DOI
10.1007/978-981-16-1754-6
Softcover ISBN
978-981-16-1753-9
Series ISSN
2191-8198
Edition Number
1
Number of Pages
XI, 121
Number of Illustrations
15 b/w illustrations
Topics