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Optimal transportation under controlled stochastic dynamics

Tan, Xiaolu; Touzi, Nizar (2013), Optimal transportation under controlled stochastic dynamics, Annals of Probability, 41, 5, p. 3201-3240. http://dx.doi.org/10.1214/12-AOP797

Type
Article accepté pour publication ou publié
Date
2013
Journal name
Annals of Probability
Volume
41
Number
5
Publisher
AIMS
Pages
3201-3240
Publication identifier
http://dx.doi.org/10.1214/12-AOP797
Metadata
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Author(s)
Tan, Xiaolu
Touzi, Nizar
Abstract (EN)
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.
Subjects / Keywords
gradient projection algorithm; viscosity solutions; Kantorovitch duality; Mass transportation

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