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Stochastic Optimal Transportation = Stochastic Control with Fixed Marginals /

紀錄類型:	書目-語言資料,印刷品 : Monograph/item
正題名/作者:	Stochastic Optimal Transportation/ by Toshio Mikami.
其他題名:	Stochastic Control with Fixed Marginals /
作者:	Mikami, Toshio.
面頁冊數:	XI, 121 p. 15 illus.online resource. :
Contained By:	Springer Nature eBook
標題:	Measure and Integration. -
電子資源:	https://doi.org/10.1007/978-981-16-1754-6
ISBN:	9789811617546

Stochastic Optimal Transportation = Stochastic Control with Fixed Marginals /
Mikami, Toshio.

Stochastic Optimal Transportation Stochastic Control with Fixed Marginals /[electronic resource] :by Toshio Mikami. - 1st ed. 2021. - XI, 121 p. 15 illus.online resource. - SpringerBriefs in Mathematics,2191-8201. - SpringerBriefs in Mathematics,.

Chapter 1. Introduction -- Chapter 2. Stochastic optimal transportation problem -- Chapter 3. Marginal problem.

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.

ISBN: 9789811617546

Standard No.: 10.1007/978-981-16-1754-6doiSubjects--Topical Terms:

672015
Measure and Integration.

LC Class. No.: QA273.A1-274.9

Dewey Class. No.: 519.2

Stochastic Optimal Transportation = Stochastic Control with Fixed Marginals /
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