The Annals of Probability

An iterated Azéma–Yor type embedding for finitely many marginals

Jan Obłój and Peter Spoida

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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.

Article information

Ann. Probab., Volume 45, Number 4 (2017), 2210-2247.

Received: February 2014
Revised: October 2015
First available in Project Euclid: 11 August 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60G44: Martingales with continuous parameter

Skorokhod embedding problem Azéma–Yor embedding maximum process martingale optimal transport continuous martingale marginal constraints


Obłój, Jan; Spoida, Peter. An iterated Azéma–Yor type embedding for finitely many marginals. Ann. Probab. 45 (2017), no. 4, 2210--2247. doi:10.1214/16-AOP1110.

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