×

Fine properties of the optimal Skorokhod embedding problem. (English) Zbl 1504.60062

The authors study the problem of stopping of a Brownian motion at a given distribution by the optimisation of a reward function that depends on the possibly randomised stopping time and the Brownian motion. The first result establishes the fact that the set \(\mathcal{T}(\nu)\) of stopping times embedding \(\nu\) is weakly dense in the set \(\mathcal{R}(\nu)\) of randomised embeddings. In particular, the optimal Skorokhod embedding problem over \(\mathcal{T}(\nu)\) has the same value as the relaxed one over \(\mathcal{R}(\nu)\), when the reward function is semicontinuous. The latter stays in parallel to a fundamental result about Monge maps and Kantorovich couplings in optimal transport. The second part studies the dual optimisation in the sense of linear programming. While existence of a dual solution failed in previous formulations, the authors introduce a relaxation of the dual problem that exploits a novel compactness property and yields existence of solutions as well as absence of a duality gap, even for irregular reward functions. This leads to a monotonicity principle which complements the key theorem of [M. Beiglböck et al., Invent. Math. 208, No. 2, 327–400 (2017; Zbl 1371.60072)]. The authors also show that the obtained results can be applied for the characterisation of the geometry of optimal embeddings through a variational condition.

MSC:

60G40 Stopping times; optimal stopping problems; gambling theory
60G44 Martingales with continuous parameter
90C08 Special problems of linear programming (transportation, multi-index, data envelopment analysis, etc.)

Citations:

Zbl 1371.60072
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Acciaio, B., Beiglböck, M., Penkner, F., Schachermayer, W.: A model-free version of the fundamental theorem of asset pricing and the super-replication theorem. Math. Finance 26, 233-251 (2016) Zbl 1378.91129 MR 3481303 · Zbl 1378.91129
[2] Aliprantis, C. D., Border, K. C.: Infinite Dimensional Analysis. 3rd ed., Springer, Berlin (2006) Zbl 1156.46001 MR 2378491 · Zbl 1156.46001
[3] Ambrosio, L.: Lecture notes on optimal transport problems. In: Mathematical Aspects of Evolving Interfaces (Funchal, 2000), Lecture Notes in Math. 1812, Springer, Berlin, 1-52 (2003) Zbl 1047.35001 MR 2011032 · Zbl 1047.35001
[4] Ambrosio, L., Gigli, N.: A user’s guide to optimal transport. In: Modelling and Optimisation of Flows on Networks, Lecture Notes in Math. 2062, Springer, Heidelberg, 1-155 (2013) MR 3050280
[5] Baxter, J. R., Chacon, R. V.: Compactness of stopping times. Z. Wahrsch. Verw. Gebiete 40, 169-181 (1977) Zbl 0349.60048 MR 517871 · Zbl 0349.60048
[6] Beiglböck, M., Cox, A. M. G., Huesmann, M.: Optimal transport and Skorokhod embedding. Invent. Math. 208, 327-400 (2017) Zbl 371.60072 MR 3639595 · Zbl 1371.60072
[7] Beiglböck, M., Cox, A. M. G., Huesmann, M.: The geometry of multi-marginal Skorokhod embedding. Probab. Theory Related Fields 176, 1045-1096 (2020) Zbl 07191239 MR 4087489 · Zbl 1469.60126
[8] Beiglböck, M., Henry-Labordère, P., Penkner, F.: Model-independent bounds for option prices-a mass transport approach. Finance Stoch. 17, 477-501 (2013) Zbl 1277.91162 MR 3066985 · Zbl 1277.91162
[9] Beiglböck, M., Henry-Labordère, P., Touzi, N.: Monotone martingale transport plans and Sko-rokhod embedding. Stochastic Process. Appl. 127, 3005-3013 (2017) Zbl 1372.60059 MR 3682121 · Zbl 1372.60059
[10] Beiglböck, M., Huesmann, M., Stebegg, F.: Root to Kellerer. In: Séminaire de probabilités XLVIII, Lecture Notes in Math. 2168, Springer, Berlin, 1-12 (2016) Zbl 1370.60083 MR 3618124 · Zbl 1370.60083
[11] Beiglböck, M., Lim, T., Obłój, J.: Dual attainment for the martingale transport problem. Bernoulli 25, 1640-1658 (2019) Zbl 07066234 MR 3961225 · Zbl 1470.49071
[12] Beiglböck, M., Nutz, M., Touzi, N.: Complete duality for martingale optimal transport on the line. Ann. Probab. 45, 3038-3074 (2017) Zbl 1417.60032 MR 3706738 · Zbl 1417.60032
[13] Bertsekas, D. P., Shreve, S. E.: Stochastic Optimal Control. Math. Sci. Engrg. 139, Academic Press, New York (1978) Zbl 0471.93002 MR 511544 · Zbl 0471.93002
[14] Biagini, S., Bouchard, B., Kardaras, C., Nutz, M.: Robust fundamental theorem for continuous processes. Math. Finance 27, 963-987 (2017) Zbl 1411.91543 MR 3705159 · Zbl 1411.91543
[15] Bouchard, B., Nutz, M.: Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab. 25, 823-859 (2015) Zbl 1322.60045 MR 3313756 · Zbl 1322.60045
[16] Brown, H., Hobson, D., Rogers, L. C. G.: Robust hedging of barrier options. Math. Finance 11, 285-314 (2001) Zbl 1047.91024 MR 1839367 · Zbl 1047.91024
[17] Bruggeman, C., Ruf, J.: A one-dimensional diffusion hits points fast. Electron. Comm. Probab. 21, art. 22, 7 pp. (2016) Zbl 1338.60197 MR 3485391 · Zbl 1338.60197
[18] Burzoni, M., Frittelli, M., Maggis, M.: Model-free superhedging duality. Ann. Appl. Probab. 27, 1452-1477 (2017) Zbl 1370.60004 MR 3678476 · Zbl 1370.60004
[19] Cheridito, P., Kupper, M., Tangpi, L.: Representation of increasing convex functionals with countably additive measures. arXiv:1502.05763 (2015)
[20] Cox, A. M. G., Hou, Z., Obłój, J.: Robust pricing and hedging under trading restrictions and the emergence of local martingale models. Finance Stoch. 20, 669-704 (2016) Zbl 1369.91175 MR 3519165 · Zbl 1369.91175
[21] Cox, A. M. G., Kinsley, S. M.: Discretisation and duality of optimal Skorokhod embedding problems. Stochastic Process. Appl. 129, 2376-2405 (2019) Zbl 07074613 MR 3958436 · Zbl 1478.60132
[22] Cox, A. M. G., Kinsley, S. M.: Robust hedging of options on a leveraged exchange traded fund. Ann. Appl. Probab. 29, 531-576 (2019) Zbl 1409.60064 MR 3910011 · Zbl 1409.60064
[23] Cox, A. M. G., Obłój, J.: Robust pricing and hedging of double no-touch options. Finance Stoch. 15, 573-605 (2011) Zbl 1303.91171 MR 2833100 · Zbl 1303.91171
[24] Cox, A. M. G., Obłój, J., Touzi, N.: The Root solution to the multi-marginal embedding prob-lem: an optimal stopping and time-reversal approach. Probab. Theory Related Fields 173, 211-259 (2019) Zbl 07030871 MR 3916107 · Zbl 1478.60133
[25] Cox, A. M. G., Wang, J.: Root’s barrier: construction, optimality and applications to variance options. Ann. Appl. Probab. 23, 859-894 (2013) Zbl 1266.91101 MR 3076672 · Zbl 1266.91101
[26] Czichowsky, C., Schachermayer, W.: Strong supermartingales and limits of nonnegative mar-tingales. Ann. Probab. 44, 171-205 (2016) Zbl 1339.60045 MR 3456335 · Zbl 1339.60045
[27] Dalang, R. C.: Sur l’arrêt optimal de processus à temps multidimensionnel continu. In: Sem-inar on Probability, XVIII, Lecture Notes in Math. 1059, Springer, Berlin, 379-390 (1984) Zbl 0537.60035 MR 770972 · Zbl 0537.60035
[28] De Marco, S., Henry-Labordère, P.: Linking vanillas and VIX options: a constrained martin-gale optimal transport problem. SIAM J. Financial Math. 6, 1171-1194 (2015) Zbl 1386.91138 MR 3432145 · Zbl 1386.91138
[29] Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pric-ing. Math. Ann. 300, 463-520 (1994) Zbl 0865.90014 MR 1304434 · Zbl 0865.90014
[30] Dellacherie, C., Meyer, P.-A.: Probabilities and Potential. North-Holland Math. Stud. 29, North-Holland, Amsterdam (1978) Zbl 0494.60001 MR 521810 · Zbl 0494.60001
[31] Dellacherie, C., Meyer, P.-A.: Probabilities and Potential. B. North-Holland Math. Stud. 72, North-Holland, Amsterdam (1982) Zbl 0494.60001 MR 745449 · Zbl 0494.60002
[32] Dolinsky, Y., Soner, H. M.: Martingale optimal transport and robust hedging in continuous time. Probab. Theory Related Fields 160, 391-427 (2014) Zbl 1305.91215 MR 3256817 · Zbl 1305.91215
[33] Dolinsky, Y., Soner, H. M.: Martingale optimal transport in the Skorokhod space. Stochastic Process. Appl. 125, 3893-3931 (2015) Zbl 1337.91092 MR 3373308 · Zbl 1337.91092
[34] El Karoui, N., Lepeltier, J.-P., Millet, A.: A probabilistic approach to the reduite in optimal stopping. Probab. Math. Statist. 13, 97-121 (1992) Zbl 0777.60034 MR 1199792 · Zbl 0777.60034
[35] Fahim, A., Huang, Y.-J.: Model-independent superhedging under portfolio constraints. Finance Stoch. 20, 51-81 (2016) Zbl 1391.91156 MR 3441286 · Zbl 1391.91156
[36] Galichon, A., Henry-Labordère, P., Touzi, N.: A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options. Ann. Appl. Probab. 24, 312-336 (2014) Zbl 1285.49012 MR 3161649 · Zbl 1285.49012
[37] Ghoussoub, N.: An integral representation of randomized probabilities and its applications. In: Seminar on Probability, XVI, Lecture Notes in Math. 920, Springer, Berlin, 519-543 (1982) Zbl 0493.60005 MR 658713 · Zbl 0493.60005
[38] Ghoussoub, N., Kim, Y.-H., Lim, T.: Structure of optimal martingale transport plans in general dimensions. Ann. Probab. 47, 109-164 (2019) Zbl 1447.60070 MR 3909967 · Zbl 1447.60070
[39] Ghoussoub, N., Kim, Y.-H., Palmer, A. Z.: PDE methods for optimal Skorokhod embeddings. Calc. Var. Partial Differential Equations 58, art. 113, 31 pp. (2019) Zbl 1416.49019 MR 3959935 · Zbl 1416.49019
[40] Gozlan, N., Roberto, C., Samson, P.-M., Tetali, P.: Kantorovich duality for general transport costs and applications. J. Funct. Anal. 273, 3327-3405 (2017) Zbl 1406.60032 MR 3706606 · Zbl 1406.60032
[41] Guo, G., Tan, X., Touzi, N.: On the monotonicity principle of optimal Skorokhod embedding problem. SIAM J. Control Optim. 54, 2478-2489 (2016) Zbl 1348.60066 MR 3549873 · Zbl 1348.60066
[42] Guo, G., Tan, X., Touzi, N.: Optimal Skorokhod embedding under finitely many marginal constraints. SIAM J. Control Optim. 54, 2174-2201 (2016) Zbl 1351.60048 MR 3539889 · Zbl 1351.60048
[43] Gyöngy, I., Šiška, D.: On randomized stopping. Bernoulli 14, 352-361 (2008) Zbl 1157.60316 MR 2544091 · Zbl 1157.60316
[44] Henry-Labordère, P., Obłój, J., Spoida, P., Touzi, N.: The maximum maximum of a martingale with given n marginals. Ann. Appl. Probab. 26, 1-44 (2016) Zbl 1337.60078 MR 3449312 · Zbl 1337.60078
[45] Henry-Labordère, P., Tan, X., Touzi, N.: An explicit martingale version of the one-dimensional Brenier’s theorem with full marginals constraint. Stochastic Process. Appl. 126, 2800-2834 (2016) Zbl 1346.60058 MR 3522302 · Zbl 1346.60058
[46] Hirsch, F., Profeta, C., Roynette, B., Yor, M.: Peacocks and Associated Martingales, with Explicit Constructions. Bocconi & Springer Ser. 3, Springer, Milan, and Bocconi Univ. Press, Milan (2011) Zbl 1227.60001 MR 2808243 · Zbl 1227.60001
[47] Hobson, D.: Robust hedging of the lookback option. Finance Stoch. 2, 329-347 (1998) Zbl 0907.90023 · Zbl 0907.90023
[48] Hobson, D.: The Skorokhod embedding problem and model-independent bounds for option prices. In: Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Math. 2003, Springer, Berlin, 267-318 (2011) Zbl 1214.91113 MR 2762363 · Zbl 1214.91113
[49] Hobson, D.: Mimicking martingales. Ann. Appl. Probab. 26, 2273-2303 (2016) Zbl 1352.60061 MR 3543897 · Zbl 1352.60061
[50] Hobson, D., Klimmek, M.: Model-independent hedging strategies for variance swaps. Finance Stoch. 16, 611-649 (2012) Zbl 1262.91134 MR 2972236 · Zbl 1262.91134
[51] Hobson, D., Klimmek, M.: Robust price bounds for the forward starting straddle. Finance Stoch. 19, 189-214 (2015) Zbl 1396.91735 MR 3292129 · Zbl 1396.91735
[52] Hobson, D., Neuberger, A.: Robust bounds for forward start options. Math. Finance 22, 31-56 (2012) Zbl 1278.91162 MR 2881879 · Zbl 1278.91162
[53] Hobson, D. G., Pedersen, J. L.: The minimum maximum of a continuous martingale with given initial and terminal laws. Ann. Probab. 30, 978-999 (2002) Zbl 1016.60047 MR 1906424 · Zbl 1016.60047
[54] Huesmann, M., Stebegg, F.: Monotonicity preserving transformations of MOT and SEP. Stochastic Process. Appl. 128, 1114-1134 (2018) Zbl 1391.60095 MR 3769657 · Zbl 1391.60095
[55] Jacod, J., Mémin, J.: Sur un type de convergence intermédiaire entre la convergence en loi et la convergence en probabilité. In: Seminar on Probability, XV (Strasbourg, 1979/1980), Lecture Notes in Math. 850, Springer, Berlin, 529-546 (1981) Zbl 0458.60016 MR 622586 · Zbl 0458.60016
[56] Källblad, S., Tan, X., Touzi, N.: Optimal Skorokhod embedding given full marginals and Azéma-Yor peacocks. Ann. Appl. Probab. 27, 686-719 (2017) Zbl 1370.60075 MR 3655851 · Zbl 1370.60075
[57] Karatzas, I., Shreve, S. E.: Brownian Motion and Stochastic Calculus. 2nd ed., Grad. Texts in Math. 113, Springer, New York (1991) Zbl 0734.60060 MR 1121940 · Zbl 0734.60060
[58] Kellerer, H. G.: Duality theorems for marginal problems. Z. Wahrsch. Verw. Gebiete 67, 399-432 (1984) Zbl 0535.60002 MR 761565 · Zbl 0535.60002
[59] Lacker, D.: Dense sets of joint distributions appearing in filtration enlargements, stochastic control, and causal optimal transport. arXiv:1805.03185 (2018)
[60] Loynes, R. M.: Stopping times on Brownian motion: Some properties of Root’s construction. Z. Wahrsch. Verw. Gebiete 16, 211-218 (1970) Zbl 0193.45701 MR 292170 · Zbl 0193.45701
[61] Madan, D. B., Yor, M.: Making Markov martingales meet marginals: with explicit construc-tions. Bernoulli 8, 509-536 (2002) Zbl 1009.60037 MR 1914701 · Zbl 1009.60037
[62] McCann, R. J.: Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80, 309-323 (1995) Zbl 0873.28009 MR 1369395 · Zbl 0873.28009
[63] Mertens, J.-F.: Théorie des processus stochastiques généraux; applications aux surmartingales. Z. Wahrsch. Verw. Gebiete 22, 45-68 (1972) Zbl 0236.60033 MR 0346895 · Zbl 0236.60033
[64] Mertens, J.-F.: Strongly supermedian functions and optimal stopping. Z. Wahrsch. Verw. Ge-biete 26, 119-139 (1973) Zbl 0297.60038 MR 0346896 · Zbl 0297.60038
[65] Monroe, I.: On embedding right continuous martingales in Brownian motion. Ann. Math. Statist. 43, 1293-1311 (1972) Zbl 0267.60050 MR 343354 · Zbl 0267.60050
[66] Neufeld, A., Nutz, M.: Superreplication under volatility uncertainty for measurable claims. Electron. J. Probab. 18, art. 48, 14 pp. (2013) Zbl 1282.91360 MR 3048120 · Zbl 1282.91360
[67] Nutz, M.: Superreplication under model uncertainty in discrete time. Finance Stoch. 18, 791-803 (2014) Zbl 1312.60049 MR 3255751 · Zbl 1312.60049
[68] Nutz, M.: Robust superhedging with jumps and diffusion. Stochastic Process. Appl. 125, 4543-4555 (2015) Zbl 1326.60120 MR 3406595 · Zbl 1326.60120
[69] Nutz, M., Stebegg, F.: Canonical supermartingale couplings. Ann. Probab. 46, 3351-3398 (2018) Zbl 1435.60030 MR 3857858 · Zbl 1435.60030
[70] Nutz, M., Stebegg, F., Tan, X.: Multiperiod martingale transport. Stochastic Process. Appl. 130, 1568-1615 (2020) Zbl 1444.60033 MR 4058283 · Zbl 1444.60033
[71] Obłój, J.: The Skorokhod embedding problem and its offspring. Probab. Surv. 1, 321-390 (2004) Zbl 1189.60088 MR 2068476 · Zbl 1189.60088
[72] Parthasarathy, K. R.: Probability Measures on Metric Spaces. Academic Press, New York (1967) Zbl 0153.19101 MR 0226684 · Zbl 0153.19101
[73] Pratelli, A.: On the equality between Monge’s infimum and Kantorovich’s minimum in opti-mal mass transportation. Ann. Inst. H. Poincaré Probab. Statist. 43, 1-13 (2007) Zbl 1121.49036 MR 2288266 · Zbl 1121.49036
[74] Rachev, S. T., Rüschendorf, L.: Mass Transportation Problems. Vol. I, Springer, New York (1998) Zbl 0990.60500 MR 1619170 · Zbl 0990.60500
[75] Rachev, S. T., Rüschendorf, L.: Mass Transportation Problems. Vol. II, Springer, New York (1998) Zbl 0990.60500 MR 1619171 · Zbl 0990.60500
[76] Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. 3rd ed., Grundlehren Math. Wiss. 293, Springer, Berlin (1999) Zbl 0917.60006 MR 1725357 · Zbl 0917.60006
[77] Root, D. H.: The existence of certain stopping times on Brownian motion. Ann. Math. Statist. 40, 715-718 (1969) Zbl 0174.21902 MR 238394 · Zbl 0174.21902
[78] Rost, H.: Skorokhod stopping times of minimal variance. In: Séminaire de Probabilités, X (Strasbourg, 1974/1975), Lecture Notes in Math. 511, 194-208 (1976) Zbl 0339.60042 MR 0445600 · Zbl 0339.60042
[79] Skorokhod, A. V.: Studies in the Theory of Random Processes. Addison-Wesley, Reading, MA (1965) Zbl 0146.37701 MR 0185620 · Zbl 0146.37701
[80] Stebegg, F.: Model-independent pricing of Asian options via optimal martingale transport. arXiv:1412.1429 (2014)
[81] Tan, X., Touzi, N.: Optimal transportation under controlled stochastic dynamics. Ann. Probab. 41, 3201-3240 (2013) Zbl 1283.60097 MR 3127880 · Zbl 1283.60097
[82] Touzi, N.: Martingale inequalities, optimal martingale transport, and robust superhedging. In: Congrès SMAI 2013, ESAIM Proc. Surveys 45, EDP Sci., Les Ulis, 32-47 (2014) Zbl 1356.60069 MR 3451815 · Zbl 1356.60069
[83] Villani, C.: Topics in Optimal Transportation. Grad. Stud. Math. 58, Amer. Math. Soc., Prov-idence, RI (2003) Zbl 1106.90001 MR 1964483 · Zbl 1106.90001
[84] Villani, C.: Optimal Transport. Grundlehren Math, Wiss. 338, Springer, Berlin (2009) Zbl 1156.53003 MR 2459454 · Zbl 1156.53003
[85] Zaev, D. A.: On the Monge-Kantorovich problem with additional linear constraints. Mat. Zametki 98, 664-683 (2015) (in Russian); · Zbl 1336.49056
[86] English transl.: Math. Notes 98, 725-741 (2015) Zbl 1336.49056 MR 3438523
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.