An iterated Azéma.Yor type embedding for finitely many marginals

We solve the n-marginal Skorokhod embedding problem for a continuous local martingale and a sequence of probability measures .1,.,.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.