TY - JOUR

T1 - Optimal Skorokhod embedding given full marginals and Azéma-Yor peacocks

AU - Källblad, Sigrid

AU - Tan, Xiaolu

AU - Touzi, Nizar

N1 - Funding Information:
Supported by ERC 321111 Rofirm, the ANR Isotace, the Chairs Financial Risks (Risk Foundation, sponsored by Société Générale), Finance and Sustainable Development (IEF sponsored by EDF and CA).
Publisher Copyright:
© Institute of Mathematical Statistics, 2017.

PY - 2017/4

Y1 - 2017/4

N2 - We consider the optimal Skorokhod embedding problem (SEP) given full marginals over the time interval [0, 1]. The problem is related to the study of extremal martingales associated with a peacock ("process increasing in convex order," by Hirsch, Profeta, Roynette and Yor [Peacocks and Associated Martingales, with Explicit Constructions (2011), Springer, Milan]). A general duality result is obtained by convergence techniques. We then study the case where the reward function depends on the maximum of the embedding process, which is the limit of the martingale transport problem studied in Henry-Labordère, Obłój, Spoida and Touzi [Ann. Appl. Probab. 26 (2016) 1-44]. Under technical conditions, we then characterize the optimal value and the solution to the dual problem. In particular, the optimal embedding corresponds to the Madan and Yor [Bernoulli 8 (2002) 509-536] peacock under their "increasing mean residual value" condition. We also discuss the associated martingale inequality.

AB - We consider the optimal Skorokhod embedding problem (SEP) given full marginals over the time interval [0, 1]. The problem is related to the study of extremal martingales associated with a peacock ("process increasing in convex order," by Hirsch, Profeta, Roynette and Yor [Peacocks and Associated Martingales, with Explicit Constructions (2011), Springer, Milan]). A general duality result is obtained by convergence techniques. We then study the case where the reward function depends on the maximum of the embedding process, which is the limit of the martingale transport problem studied in Henry-Labordère, Obłój, Spoida and Touzi [Ann. Appl. Probab. 26 (2016) 1-44]. Under technical conditions, we then characterize the optimal value and the solution to the dual problem. In particular, the optimal embedding corresponds to the Madan and Yor [Bernoulli 8 (2002) 509-536] peacock under their "increasing mean residual value" condition. We also discuss the associated martingale inequality.

KW - Martingale inequality

KW - Martingale transport problem

KW - Maximum of martingale given marginals

KW - Peacocks

KW - Skorokhod embedding problem

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U2 - 10.1214/16-AAP1191

DO - 10.1214/16-AAP1191

M3 - Article

AN - SCOPUS:85019669721

SN - 1050-5164

VL - 27

SP - 686

EP - 719

JO - Annals of Applied Probability

JF - Annals of Applied Probability

IS - 2

ER -