The Annals of Probability

Canonical supermartingale couplings

Marcel Nutz and Florian Stebegg

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Two probability distributions $\mu$ and $\nu$ in second stochastic order can be coupled by a supermartingale, and in fact by many. Is there a canonical choice? We construct and investigate two couplings which arise as optimizers for constrained Monge–Kantorovich optimal transport problems where only supermartingales are allowed as transports. Much like the Hoeffding–Fréchet coupling of classical transport and its symmetric counterpart, the antitone coupling, these can be characterized by order-theoretic minimality properties, as simultaneous optimal transports for certain classes of reward (or cost) functions, and through no-crossing conditions on their supports; however, our two couplings have asymmetric geometries. Remarkably, supermartingale optimal transport decomposes into classical and martingale transport in several ways.

Article information

Ann. Probab., Volume 46, Number 6 (2018), 3351-3398.

Received: July 2017
Revised: November 2017
First available in Project Euclid: 25 September 2018

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Mathematical Reviews number (MathSciNet)

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

Coupling optimal transport Spence–Mirrlees condition


Nutz, Marcel; Stebegg, Florian. Canonical supermartingale couplings. Ann. Probab. 46 (2018), no. 6, 3351--3398. doi:10.1214/17-AOP1249.

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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. (2017). Optimal transport and Skorokhod embedding. Invent. Math. 208 327–400.
  • [4] Beiglböck, M., Cox, A. M. G., Huesmann, M., Perkowski, N. and Prömel, D. J. (2017). Pathwise superreplication via Vovk’s outer measure. Finance Stoch. 21 1141–1166.
  • [5] Beiglböck, M., Goldstern, M., Maresch, G. and Schachermayer, W. (2009). Optimal and better transport plans. J. Funct. Anal. 256 1907–1927.
  • [6] Beiglböck, M., Henry-Labordère, P. and Penkner, F. (2013). Model-independent bounds for option prices: A mass transport approach. Finance Stoch. 17 477–501.
  • [7] Beiglböck, M., Henry-Labordère, P. and Touzi, N. (2017). Monotone martingale transport plans and Skorokhod embedding. Stochastic Process. Appl. 127 3005–3013.
  • [8] Beiglböck, M., Huesmann, M. and Stebegg, F. (2016). Root to Kellerer. In Séminaire de Probabilités XLVIII. Lecture Notes in Math. 2168 1–12. Springer, Berlin.
  • [9] Beiglböck, M. and Juillet, N. (2016). On a problem of optimal transport under marginal martingale constraints. Ann. Probab. 44 42–106.
  • [10] Beiglböck, M. and Nutz, M. (2014). Martingale inequalities and deterministic counterparts. Electron. J. Probab. 19 1–15.
  • [11] Beiglböck, M., Nutz, M. and Touzi, N. (2017). Complete duality for martingale optimal transport on the line. Ann. Probab. 45 3038–3074.
  • [12] Beiglböck, M. and Pratelli, A. (2012). Duality for rectified cost functions. Calc. Var. Partial Differential Equations 45 27–41.
  • [13] Bertsekas, D. P. and Shreve, S. E. (1978). Stochastic Optimal Control. The Discrete-Time Case. Academic Press, New York.
  • [14] Biagini, S., Bouchard, B., Kardaras, C. and Nutz, M. (2017). Robust fundamental theorem for continuous processes. Math. Finance 27 963–987.
  • [15] Bouchard, B. and Nutz, M. (2015). Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab. 25 823–859.
  • [16] Burzoni, M., Frittelli, M. and Maggis, M. (2017). Model-free superhedging duality. Ann. Appl. Probab. 27 1452–1477.
  • [17] Campi, L., Laachir, I. and Martini, C. (2017). Change of numeraire in the two-marginals martingale transport problem. Finance Stoch. 21 471–486.
  • [18] Cheridito, P., Kupper, M. and Tangpi, L. (2015). Representation of increasing convex functionals with countably additive measures. Preprint. Available at arXiv:1502.05763v1.
  • [19] Cox, A. M. G. (2008). Extending Chacon–Walsh: Minimality and generalised starting distributions. In Séminaire de Probabilités XLI. Lecture Notes in Math. 1934 233–264. Springer, Berlin.
  • [20] 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.
  • [21] Cox, A. M. G. and Obłój, J. (2011). Robust pricing and hedging of double no-touch options. Finance Stoch. 15 573–605.
  • [22] Cox, A. M. G., Obłój, J. and Touzi, N. (2015). The Root solution to the multi-marginal embedding problem: An optimal stopping and time-reversal approach. Preprint. Available at arXiv:1505.03169v1.
  • [23] De Marco, S. and Henry-Labordère, P. (2015). Linking vanillas and VIX options: A constrained martingale optimal transport problem. SIAM J. Financial Math. 6 1171–1194.
  • [24] Dolinsky, Y. and Neufeld, A. (2015). Super-replication in extremely incomplete markets. Math. Finance. To appear.
  • [25] Dolinsky, Y. and Soner, H. M. (2014). Martingale optimal transport and robust hedging in continuous time. Probab. Theory Related Fields 160 391–427.
  • [26] Dolinsky, Y. and Soner, H. M. (2015). Martingale optimal transport in the Skorokhod space. Stochastic Process. Appl. 125 3893–3931.
  • [27] Fahim, A. and Huang, Y.-J. (2016). Model-independent superhedging under portfolio constraints. Finance Stoch. 20 51–81.
  • [28] Föllmer, H. and Schied, A. (2011). Stochastic Finance: An Introduction in Discrete Time, 3rd ed. de Gruyter, Berlin.
  • [29] 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.
  • [30] Ghoussoub, N., Kim, Y.-H. and Lim, T. (2015). Structure of optimal martingale transport in general dimensions. Preprint. Available at arXiv:1508.01806v1.
  • [31] Gozlan, N., Roberto, C., Samson, P.-M. and Tetali, P. (2017). Kantorovich duality for general transport costs and applications. J. Funct. Anal. 273 3327–3405.
  • [32] Griessler, C. (2016). An extended footnote on finitely minimal martingale measures. Preprint. Available at arXiv:1606.03106v1.
  • [33] Guo, G., Tan, X. and Touzi, N. (2016). On the monotonicity principle of optimal Skorokhod embedding problem. SIAM J. Control Optim. 54 2478–2489.
  • [34] Guo, G., Tan, X. and Touzi, N. (2016). Optimal Skorokhod embedding under finitely many marginal constraints. SIAM J. Control Optim. 54 2174–2201.
  • [35] Guo, G., Tan, X. and Touzi, N. (2017). Tightness and duality of martingale transport on the Skorokhod space. Stochastic Process. Appl. 127 927–956.
  • [36] Henry-Labordère, P., Obłój, J., Spoida, P. and Touzi, N. (2016). Maximum maximum of martingales given marginals. Ann. Appl. Probab. 26 1–44.
  • [37] Henry-Labordère, P., Tan, X. and Touzi, N. (2016). An explicit version of the one-dimensional Brenier’s theorem with full marginals constraint. Stochastic Process. Appl. 126 2800–2834.
  • [38] Henry-Labordère, P. and Touzi, N. (2016). An explicit martingale version of the one-dimensional Brenier theorem. Finance Stoch. 20 635–668.
  • [39] Hirsch, F., Profeta, C., Roynette, B. and Yor, M. (2011). Peacocks and Associated Martingales, with Explicit Constructions. Bocconi & Springer Series 3. Springer, Milan; Bocconi Univ. Press, Milan.
  • [40] Hobson, D. (1998). Robust hedging of the lookback option. Finance Stoch. 2 329–347.
  • [41] 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.
  • [42] Hobson, D. (2016). Mimicking martingales. Ann. Appl. Probab. 26 2273–2303.
  • [43] Hobson, D. and Klimmek, M. (2015). Robust price bounds for the forward starting straddle. Finance Stoch. 19 189–214.
  • [44] Hobson, D. and Neuberger, A. (2012). Robust bounds for forward start options. Math. Finance 22 31–56.
  • [45] Juillet, N. (2016). Stability of the shadow projection and the left-curtain coupling. Ann. Inst. Henri Poincaré Probab. Stat. 52 1823–1843.
  • [46] Källblad, S., Tan, X. and Touzi, N. (2017). Optimal Skorokhod embedding given full marginals and Azéma–Yor peacocks. Ann. Appl. Probab. 27 686–719.
  • [47] Kechris, A. S. (1995). Classical Descriptive Set Theory. Graduate Texts in Mathematics 156. Springer, New York.
  • [48] Kellerer, H. G. (1984). Duality theorems for marginal problems. Z. Wahrsch. Verw. Gebiete 67 399–432.
  • [49] Neufeld, A. and Nutz, M. (2013). Superreplication under volatility uncertainty for measurable claims. Electron. J. Probab. 18 1–14.
  • [50] Nutz, M. (2014). Superreplication under model uncertainty in discrete time. Finance Stoch. 18 791–803.
  • [51] Nutz, M. (2015). Robust superhedging with jumps and diffusion. Stochastic Process. Appl. 125 4543–4555.
  • [52] Nutz, M., Stebegg, F. and Tan, X. (2017). Multiperiod martingale transport. Preprint. Available at arXiv:1703.10588v1.
  • [53] Obłój, J. (2004). The Skorokhod embedding problem and its offspring. Probab. Surv. 1 321–390.
  • [54] Obłój, J. and Spoida, P. (2017). An iterated Azéma–Yor type embedding for finitely many marginals. Ann. Probab. 45 2210–2247.
  • [55] Rachev, S. T. and Rüschendorf, L. (1998). Mass Transportation Problems, Vol. I: Theory. Springer, New York.
  • [56] Rachev, S. T. and Rüschendorf, L. (1998). Mass Transportation Problems, Vol. II: Applications. Springer, New York.
  • [57] Stebegg, F. (2014). Model-independent pricing of Asian options via optimal martingale transport. Preprint. Available at arXiv:1412.1429v1.
  • [58] Tan, X. and Touzi, N. (2013). Optimal transportation under controlled stochastic dynamics. Ann. Probab. 41 3201–3240.
  • [59] Touzi, N. (2014). Martingale inequalities, optimal martingale transport, and robust superhedging. In Congrès SMAI 2013. ESAIM Proc. Surveys 45 32–47. EDP Sci., Les Ulis.
  • [60] Villani, C. (2003). Topics in Optimal Transportation. Graduate Studies in Mathematics 58. Amer. Math. Soc., Providence, RI.
  • [61] Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer, Berlin.
  • [62] Zaev, D. (2015). On the Monge–Kantorovich problem with additional linear constraints. Math. Notes 98 725–741.