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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Integral equations
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Stochastic Optimal Transportation - Stochastic Control with Fixed Marginals (Paperback, 1st ed. 2021)
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Stochastic Optimal Transportation - Stochastic Control with Fixed Marginals (Paperback, 1st ed. 2021)
Series: SpringerBriefs in Mathematics
Expected to ship within 10 - 15 working days
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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 Schroedinger'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 Schroedinger'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 Schroedinger'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
Schroedinger's functional equation when the space of Borel
probability measures is endowed with a strong or a weak topology,
and it is shown that Schroedinger'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.
General
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