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Field name	Details
Dewey Class	519.2
Title	Stochastic Optimal Transportation ([EBook] :) : Stochastic Control with Fixed Marginals / / by Toshio Mikami.
Author	Mikami, Toshio
Other name(s)	SpringerLink (Online service)
Edition statement	1st ed. 2021.
Publication	Singapore : : Springer Singapore : : Imprint: Springer, , 2021.
Physical Details	XI, 121 p. 15 illus. : online resource.
Series	SpringerBriefs in Mathematics 2191-8201
ISBN	9789811617546
Summary Note	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.:
Contents note	Chapter 1. Introduction -- Chapter 2. Stochastic optimal transportation problem -- Chapter 3. Marginal problem.
Mode of acces to digital resource	Mode of access: World Wide Web. System requirements: Internet Explorer 6.0 (or higher) or Firefox 2.0 (or higher). Available as searchable text in PDF format.
System details note	Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users).
Internet Site	https://doi.org/10.1007/978-981-16-1754-6
Links to Related Works	Subject References: Differential Equations . Differential Geometry . Functional Analysis . Geometry, Differential . Measure and Integration . Measure theory . Probabilities . Probability Theory . Authors: author . Mikami, Toshio . Corporate Authors: SpringerLink (Online service) . Series: SpringerBriefs in Mathematics . Classification: 519.2 .

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Stochastic Optimal Transportation: Stochastic Control with Fixed Marginals /

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