An iterated Azéma–Yor type embedding for finitely many marginals

• Jan Obłój ^[1] ; Peter Spoida ^[1]
  1. [1] University of Oxford

     University of Oxford

     Oxford District, Reino Unido

     
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 4, 2017, págs. 2210-2247
• Idioma: inglés
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• Resumen

  • We solve the nn-marginal Skorokhod embedding problem for a continuous local martingale and a sequence of probability measures μ1,…,μnμ1,…,μn 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=3n=3.

    In our analysis we compute the law of the maximum at each of the nn 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 nn-marginal Skorokhod embedding problem. The result has direct implications for robust pricing and hedging of Lookback options.