The Annals of Probability

Complete duality for martingale optimal transport on the line

Mathias Beiglböck, Marcel Nutz, and Nizar Touzi

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We study the optimal transport between two probability measures on the real line, where the transport plans are laws of one-step martingales. A quasi-sure formulation of the dual problem is introduced and shown to yield a complete duality theory for general marginals and measurable reward (cost) functions: absence of a duality gap and existence of dual optimizers. Both properties are shown to fail in the classical formulation. As a consequence of the duality result, we obtain a general principle of cyclical monotonicity describing the geometry of optimal transports.

Article information

Ann. Probab., Volume 45, Number 5 (2017), 3038-3074.

Received: July 2015
Revised: May 2016
First available in Project Euclid: 23 September 2017

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Zentralblatt MATH identifier

Primary: 60G42: Martingales with discrete parameter 49N05: Linear optimal control problems [See also 93C05]

Martingale optimal transport Kantorovich duality


Beiglböck, Mathias; Nutz, Marcel; Touzi, Nizar. Complete duality for martingale optimal transport on the line. Ann. Probab. 45 (2017), no. 5, 3038--3074. doi:10.1214/16-AOP1131.

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  • [1] Acciaio, B., Beiglböck, M., Penkner, F. and Schachermayer, W. (2016). A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. Math. Finance 26 233–251.
  • [2] Ambrosio, L. and Gigli, N. (2013). A user’s guide to optimal transport. In Modelling and Optimisation of Flows on Networks. Lecture Notes in Math. 2062 1–155. Springer, Heidelberg.
  • [3] Beiglböck, M., Cox, A. M. G. and Huesmann, M. (2014). Optimal transport and Skorokhod embedding. Preprint. Available at arXiv:1307.3656v1.
  • [4] Beiglböck, M., Goldstern, M., Maresch, G. and Schachermayer, W. (2009). Optimal and better transport plans. J. Funct. Anal. 256 1907–1927.
  • [5] Beiglböck, M., Henry-Labordère, P. and Penkner, F. (2013). Model-independent bounds for option prices—A mass transport approach. Finance Stoch. 17 477–501.
  • [6] Beiglböck, M., Henry-Labordère, P. and Touzi, N. (2015). Monotone martingale transport plans and Skorohod embedding. Preprint.
  • [7] Beiglböck, M. and Juillet, N. (2016). On a problem of optimal transport under marginal martingale constraints. Ann. Probab. 44 42–106.
  • [8] Beiglböck, M. and Nutz, M. (2014). Martingale inequalities and deterministic counterparts. Electron. J. Probab. 19 1–15.
  • [9] Beiglböck, M. and Pratelli, A. (2012). Duality for rectified cost functions. Calc. Var. Partial Differential Equations 45 27–41.
  • [10] Bertsekas, D. P. and Shreve, S. E. (1978). Stochastic Optimal Control: The Discrete Time Case. Mathematics in Science and Engineering 139. Academic Press, New York.
  • [11] Biagini, S., Bouchard, B., Kardaras, C. and Nutz, M. (2016). Robust fundamental theorem for continuous processes. Math. Finance. To appear.
  • [12] Bogachev, V. I. (2007). Measure Theory. Vol. I. Springer, Berlin.
  • [13] Bouchard, B. and Nutz, M. (2015). Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab. 25 823–859.
  • [14] Burzoni, M., Frittelli, M. and Maggis, M. (2015). Model-free superhedging duality. Preprint. Available at arXiv:1506.06608v2.
  • [15] Campi, L., Laachir, I. and Martini, C. (2014). Change of numeraire in the two-marginals martingale transport problem. Preprint. Available at arXiv:1406.6951v3.
  • [16] Cheridito, P., Kupper, M. and Tangpi, L. (2015). Representation of increasing convex functionals with countably additive measures. Preprint. Available at arXiv:1502.05763v1.
  • [17] Cox, A. M. G., Hou, Z. and Obłój, J. (2016). Robust pricing and hedging under trading restrictions and the emergence of local martingale models. Finance Stoch. 20 669–704.
  • [18] Cox, A. M. G. and Obłój, J. (2011). Robust pricing and hedging of double no-touch options. Finance Stoch. 15 573–605.
  • [19] Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300 463–520.
  • [20] Dolinsky, Y. and Soner, H. M. (2014). Martingale optimal transport and robust hedging in continuous time. Probab. Theory Related Fields 160 391–427.
  • [21] Dolinsky, Y. and Soner, H. M. (2015). Martingale optimal transport in the Skorokhod space. Stochastic Process. Appl. 125 3893–3931.
  • [22] Fahim, A. and Huang, Y.-J. (2016). Model-independent superhedging under portfolio constraints. Finance Stoch. 20 51–81.
  • [23] Galichon, A., Henry-Labordère, P. and Touzi, N. (2014). A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options. Ann. Appl. Probab. 24 312–336.
  • [24] Henry-Labordère, P., Obłój, J., Spoida, P. and Touzi, N. (2016). The maximum maximum of a martingale with given $n$ marginals. Ann. Appl. Probab. 26 1–44.
  • [25] Henry-Labordère, P., Tan, X. and Touzi, N. (2014). A numerical algorithm for a class of BSDEs via the branching process. Stochastic Process. Appl. 124 1112–1140.
  • [26] Henry-Labordère, P. and Touzi, N. (2016). An explicit martingale version of the one-dimensional Brenier theorem. Finance Stoch. 20 635–668.
  • [27] Hobson, D. (1998). Robust hedging of the lookback option. Finance Stoch. 2 329–347.
  • [28] Hobson, D. (2011). The Skorokhod embedding problem and model-independent bounds for option prices. In Paris–Princeton Lectures on Mathematical Finance 2010. Lecture Notes in Math. 2003 267–318. Springer, Berlin.
  • [29] Hobson, D. and Klimmek, M. (2015). Robust price bounds for the forward starting straddle. Finance Stoch. 19 189–214.
  • [30] Hobson, D. and Neuberger, A. (2012). Robust bounds for forward start options. Math. Finance 22 31–56.
  • [31] Juillet, N. (2016). Stability of the shadow projection and the left-curtain coupling. Ann. Inst. Henri Poincaré Probab. Stat. 52 1823–1843.
  • [32] Källblad, S., Tan, X. and Touzi, N. (2016). Optimal Skorokhod embedding given full marginals and Azéma–Yor peacocks. Ann. Appl. Probab. To appear.
  • [33] Kellerer, H. G. (1984). Duality theorems for marginal problems. Z. Wahrsch. Verw. Gebiete 67 399–432.
  • [34] Neufeld, A. and Nutz, M. (2013). Superreplication under volatility uncertainty for measurable claims. Electron. J. Probab. 18 1–14.
  • [35] Nutz, M. (2014). Superreplication under model uncertainty in discrete time. Finance Stoch. 18 791–803.
  • [36] Nutz, M. (2015). Robust superhedging with jumps and diffusion. Stochastic Process. Appl. 125 4543–4555.
  • [37] Obłój, J. (2004). The Skorokhod embedding problem and its offspring. Probab. Surv. 1 321–390.
  • [38] Stebegg, F. (2014). Model-independent pricing of Asian options via optimal martingale transport. Preprint. Available at arXiv:1412.1429v1.
  • [39] Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Stat. 36 423–439.
  • [40] Tan, X. and Touzi, N. (2013). Optimal transportation under controlled stochastic dynamics. Ann. Probab. 41 3201–3240.
  • [41] Villani, C. (2003). Topics in Optimal Transportation. Graduate Studies in Mathematics 58. Amer. Math. Soc., Providence, RI.
  • [42] Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften 338. Springer, Berlin.
  • [43] Zaev, D. A. (2015). On the Monge–Kantorovich problem with additional linear constraints. Math. Notes 98 725–741.