Stochastic Optimal Transportation: Stochastic Control with Fixed Marginals

The optimal transportation problem (OT) is a variational problem for absolutely continuous stochastic processes with fixed initial and terminal distributions, while Schrödingers problem is a variational problem for one-step random walks with fixed initial and terminal distributions. The stochastic optimal transportation problem (SOT) is a generalization of the OT, and the recent development of the superposition principle plays a crucial role in it. The SOT is discussed in terms of duality theorem, forward–backward stochastic differential equations, and probabilistic proofs. It also discusses the regularity result for the solution to Schrödingers functional equation and the construction of stochastic processes with given marginals.

Format: Paperback / softback
Length: 121 pages
Publication date: 16 June 2021
Publisher: Springer Verlag, Singapore

The optimal transportation problem (OT) is a fascinating mathematical concept that involves the study of absolutely continuous stochastic processes with fixed initial and terminal distributions. This problem is closely related to Schrödingers problem, which is originally a variational problem for one-step random walks with fixed initial and terminal distributions. To extend the scope of OT, the stochastic optimal transportation problem (SOT) is introduced as a generalization, where semimartingales with fixed initial and terminal distributions are considered.

An interpretation of SOT is also provided, which can be seen as a generalization of Schrödingers problem. In this book, we delve into the fundamental results of SOT, including the duality theorem, a sufficient condition for the problem to be finite, forward-backward stochastic differential equations (SDEs) for the minimizer, and more.

One of the key developments in SOT is the superposition principle, which plays a crucial role in understanding and solving the problem. A systematic method is introduced to consider two problems: one with fixed initial and terminal distributions and another with fixed marginal distributions for all times. By applying the zero-noise limit of SOT, we obtain probabilistic proofs to Monges problem with a quadratic cost and the duality theorem for OT. Additionally, we explore the Lipschitz continuity and semiconcavity of Schrödingers problem in marginal distributions and random variables with given marginals, respectively.

Furthermore, we discuss the regularity result for the solution to Schrödingers functional equation when the space of Borel probability measures is endowed with a strong or a weak topology. It is shown that Schrödingers problem can be considered a class of mean field games, and the construction of stochastic processes with given margin is discussed.

Overall, this book provides a comprehensive and in-depth exploration of the optimal transportation problem and its generalizations, including the stochastic optimal transportation problem. It offers valuable insights into the mathematical principles and applications of these problems, making it a valuable resource for researchers and practitioners in the field.

Weight: 454g
Dimension: 235 x 155 (mm)
ISBN-13: 9789811617539
Edition number: 1st ed. 2021