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.
