8,335 research outputs found
Optimal transportation under controlled stochastic dynamics
We consider an extension of the Monge-Kantorovitch optimal transportation
problem. The mass is transported along a continuous semimartingale, and the
cost of transportation depends on the drift and the diffusion coefficients of
the continuous semimartingale. The optimal transportation problem minimizes the
cost among all continuous semimartingales with given initial and terminal
distributions. Our first main result is an extension of the Kantorovitch
duality to this context. We also suggest a finite-difference scheme combined
with the gradient projection algorithm to approximate the dual value. We prove
the convergence of the scheme, and we derive a rate of convergence. We finally
provide an application in the context of financial mathematics, which
originally motivated our extension of the Monge-Kantorovitch problem. Namely,
we implement our scheme to approximate no-arbitrage bounds on the prices of
exotic options given the implied volatility curve of some maturity.Comment: Published in at http://dx.doi.org/10.1214/12-AOP797 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Scaling Algorithms for Unbalanced Transport Problems
This article introduces a new class of fast algorithms to approximate
variational problems involving unbalanced optimal transport. While classical
optimal transport considers only normalized probability distributions, it is
important for many applications to be able to compute some sort of relaxed
transportation between arbitrary positive measures. A generic class of such
"unbalanced" optimal transport problems has been recently proposed by several
authors. In this paper, we show how to extend the, now classical, entropic
regularization scheme to these unbalanced problems. This gives rise to fast,
highly parallelizable algorithms that operate by performing only diagonal
scaling (i.e. pointwise multiplications) of the transportation couplings. They
are generalizations of the celebrated Sinkhorn algorithm. We show how these
methods can be used to solve unbalanced transport, unbalanced gradient flows,
and to compute unbalanced barycenters. We showcase applications to 2-D shape
modification, color transfer, and growth models
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