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July 2017 An iterated Azéma–Yor type embedding for finitely many marginals
Jan Obłój, Peter Spoida
Ann. Probab. 45(4): 2210-2247 (July 2017). DOI: 10.1214/16-AOP1110


We solve the $n$-marginal Skorokhod embedding problem for a continuous local martingale and a sequence of probability measures $\mu_{1},\ldots,\mu_{n}$ which are in convex order and satisfy an additional technical assumption. Our construction is explicit and is a multiple marginal generalization of the Azéma and Yor [In Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78) (1979) 90–115 Springer] solution. In particular, we recover the stopping boundaries obtained by Brown, Hobson and Rogers [Probab. Theory Related Fields 119 (2001) 558–578] and Madan and Yor [Bernoulli 8 (2002) 509–536]. Our technical assumption is necessary for the explicit embedding, as demonstrated with a counterexample. We discuss extensions to the general case giving details when $n=3$.

In our analysis we compute the law of the maximum at each of the $n$ stopping times. This is used in Henry-Labordère et al. [Ann. Appl. Probab. 26 (2016) 1–44] to show that the construction maximizes the distribution of the maximum among all solutions to the $n$-marginal Skorokhod embedding problem. The result has direct implications for robust pricing and hedging of Lookback options.


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Jan Obłój. Peter Spoida. "An iterated Azéma–Yor type embedding for finitely many marginals." Ann. Probab. 45 (4) 2210 - 2247, July 2017.


Received: 1 February 2014; Revised: 1 October 2015; Published: July 2017
First available in Project Euclid: 11 August 2017

zbMATH: 1380.60048
MathSciNet: MR3693961
Digital Object Identifier: 10.1214/16-AOP1110

Primary: 60G40 , 60G44

Keywords: Azéma–Yor embedding , continuous martingale , marginal constraints , Martingale optimal transport , maximum process , Skorokhod embedding problem

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 4 • July 2017
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