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September 2013 Optimal transportation under controlled stochastic dynamics
Xiaolu Tan, Nizar Touzi
Ann. Probab. 41(5): 3201-3240 (September 2013). DOI: 10.1214/12-AOP797

Abstract

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.

Citation

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Xiaolu Tan. Nizar Touzi. "Optimal transportation under controlled stochastic dynamics." Ann. Probab. 41 (5) 3201 - 3240, September 2013. https://doi.org/10.1214/12-AOP797

Information

Published: September 2013
First available in Project Euclid: 12 September 2013

zbMATH: 1283.60097
MathSciNet: MR3127880
Digital Object Identifier: 10.1214/12-AOP797

Subjects:
Primary: 60H30, 65K99
Secondary: 65P99

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 5 • September 2013
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