Stochastic Optimal Transportation

Toshio Mikami

ISBN 10: 9811617538 / ISBN 13: 9789811617539

Verlag: Springer Nature Singapore, Springer Nature Singapore Jun 2021, 2021

Beschreibung

Neuware -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 introducedto 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.Springer Verlag GmbH, Tiergartenstr. 17, 69121 Heidelberg 136 pp. Englisch.

Bestandsnummer des Verkäufers 9789811617539

Bibliografische Details

• Titel: Stochastic Optimal Transportation
• Verlag: Springer Nature Singapore, Springer Nature Singapore Jun 2021
• Erscheinungsjahr: 2021
• Sprache: Englisch
• ISBN-10: 9811617538
• ISBN-13: 9789811617539
• Einband: Taschenbuch
• Zustand: Neu

Über diesen Titel

Inhaltsangabe

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.

Über die Autorin bzw. den Autor

Charlotte y Peter Fiell son dos autoridades en historia, teoría y crítica del diseño y han escrito más de sesenta libros sobre la materia, muchos de los cuales se han convertido en éxitos de ventas. También han impartido conferencias y cursos como profesores invitados, han comisariado exposiciones y asesorado a fabricantes, museos, salas de subastas y grandes coleccionistas privados de todo el mundo. Los Fiell han escrito numerosos libros para TASCHEN, entre los que se incluyen 1000 Chairs, Diseño del siglo XX, El diseño industrial de la A a la Z, Scandinavian Design y Diseño del siglo XXI.