×

Causal optimal transport and its links to enlargement of filtrations and continuous-time stochastic optimization. (English) Zbl 1435.90012

Summary: The martingale part in the semimartingale decomposition of a Brownian motion with respect to an enlargement of its filtration, is an anticipative mapping of the given Brownian motion. In analogy to optimal transport theory, we define causal transport plans in the context of enlargement of filtrations, as the Kantorovich counterparts of the aforementioned non-adapted mappings. We provide a necessary and sufficient condition for a Brownian motion to remain a semimartingale in an enlarged filtration, in terms of certain minimization problems over sets of causal transport plans. The latter are also used in order to give robust transport-based estimates for the value of having additional information, as well as model sensitivity with respect to the reference measure, for the classical stochastic optimization problems of utility maximization and optimal stopping.

MSC:

90B06 Transportation, logistics and supply chain management
60J65 Brownian motion
90C15 Stochastic programming
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] Acciaio, B.; Backhoff-Veraguas, J.; Carmona, R., Extended mean field control problems: stochastic maximum principle and transport perspective, SIAM J. Control Optim. (2018), in press, arXiv:1802.05754
[2] Aksamit, A.; Jeanblanc, M., Enlargement of Filtration with Finance in View (2017), Springer · Zbl 1397.91003
[3] Aksamit, A.; Li, L., Projections, pseudo-stopping times and the immersion property, (Séminaire de Probabilités XLVIII (2016), Springer), 459-467 · Zbl 1367.60041
[4] D. Aldous, Weak convergence and the general theory of processes (weak convergence of stochastic processes for processes viewed in the Strasbourg manner), unpublished. 1981.
[5] Amendinger, J.; Imkeller, P.; Schweizer, M., Additional logarithmic utility of an insider, Stochastic Process. Appl., 75, 2, 263-286 (1998) · Zbl 0934.91020
[6] Ankirchner, S.; Dereich, S.; Imkeller, P., The Shannon information of filtrations and the additional logarithmic utility of insiders, Ann. Probab., 34, 2, 743-778 (2006), MR 2223957 · Zbl 1098.60065
[7] Ankirchner, S.; Dereich, S.; Imkeller, P., Enlargement of filtrations and continuous Girsanov-type embeddings, (Séminaire de Probabilités XL. Séminaire de Probabilités XL, Lecture Notes in Math., vol. 1899 (2007), Springer: Springer Berlin), 389-410, MR 2409018 · Zbl 1155.60017
[8] Backhoff-Veraguas, J.; Beiglbock, M.; Lin, Y.; Zalashko, A., Causal transport in discrete time and applications, SIAM J. Optim., 27, 4, 2528-2562 (2017) · Zbl 1387.90168
[9] Baxter, J. R.; Chacon, R. V., Compactness of stopping times, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 40, 3, 169-181 (1977), MR 0517871 · Zbl 0349.60048
[10] Beiglböck, M.; Cox, A. M.G.; Huesmann, M., Optimal transport and Skorokhod embedding, Invent. Math., 208, 2, 327-400 (2017) · Zbl 1371.60072
[11] M. Beiglböck, C. Griessler, A land of monotone plenty, no. 1, 109-127, Annali della SNS. · Zbl 1425.60043
[12] Beiglböck, M.; Henry-Labordére, P.; Penkner, F., Model-independent bounds for option prices – a mass transport approach, Finance Stoch., 17, 3, 477-501 (2013) · Zbl 1277.91162
[13] Beiglböck, M.; Schachermayer, W., Duality for Borel measurable cost functions, Trans. Amer. Math. Soc., 363, 8, 4203-4224 (2011), MR 2792985 (2012k:49108) · Zbl 1228.49046
[14] Benamou, J.-D.; Carlier, G.; Cuturi, M.; Nenna, L.; Peyré, G., Iterative Bregman projections for regularized transportation problems, SIAM J. Sci. Comput., 37, 2, A1111-A1138 (2015) · Zbl 1319.49073
[15] Bion-Nadal, J.; Talay, D., On a Wasserstein-type distance between solutions to stochastic differential equations, Ann. Appl. Probab., 29, 3, 1609-1639 (2019) · Zbl 1412.60115
[16] Brémaud, P.; Yor, M., Changes of filtrations and of probability measures, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 45, 4, 269-295 (1978) · Zbl 0415.60048
[17] D. Coculescu, M. Jeanblanc, Structure conditions under short-sales constraints and applications to converging asset prices. · Zbl 1411.91248
[18] Coquet, F.; Toldo, S., Convergence of values in optimal stopping and convergence of optimal stopping times, Electron. J. Probab., 12, 8, 207-228 (2007), MR 2299917 · Zbl 1144.62067
[19] Cuturi, M., Sinkhorn distances: Lightspeed computation of optimal transport, (Advances in Neural Information Processing Systems (2013)), 2292-2300
[20] Cvitanić, J.; Karatzas, I., Convex duality in constrained portfolio optimization, Ann. Appl. Probab., 2, 4, 767-818 (1992), MR 1189418 · Zbl 0770.90002
[21] Dellacherie, C.; Meyer, P.-A., (Probabilités et Potentiel. Chapitres V à VIII. Probabilités et Potentiel. Chapitres V à VIII, Actualités Scientifiques et Industrielles, vol. 1385 (1980), Hermann: Hermann Paris), Théorie des martingales. [Martingale theory]. MR 566768 · Zbl 0464.60001
[22] Eckstein, S.; Kupper, M., Computation of optimal transport and related hedging problems via penalization and neural networks, Appl. Math. Optim., 1-29 (2018)
[23] Feyel, D.; Üstünel, A. S., Monge-Kantorovitch measure transportation and Monge-Ampère equation on Wiener space, Probab. Theory Related Fields, 128, 3, 347-385 (2004), MR 2036490 (2004m:60121) · Zbl 1055.60052
[24] Florens, J. P.; Fougere, D., Noncausality in continuous time, Econometrica, 1195-1212 (1996) · Zbl 0856.90020
[25] Galichon, A.; Henry-Labordére, P.; Touzi, N., A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options, Ann. Appl. Probab., 24, 1, 312-336 (2014) · Zbl 1285.49012
[26] Jacod, J., Weak and strong solutions of stochastic differential equations, Stochastics, 3, 171-191 (1980) · Zbl 0434.60061
[27] Jacod, J., Grossissement initial, hypothèse (H’), et théorème de Girsanov, (Jeulin, T.; Yor, M., Grossissements de Filtrations: Exemples et Applications. Grossissements de Filtrations: Exemples et Applications, Lecture Notes in Mathematics, vol. 1118 (1985), Springer: Springer Berlin - Heidelberg), 15-35 · Zbl 0568.60049
[28] Jeanblanc, M.; Yor, M.; Chesney, M., Mathematical Methods for Financial Markets (2009), Springer Finance, Springer-Verlag London Ltd.: Springer Finance, Springer-Verlag London Ltd. London, MR 2568861 · Zbl 1205.91003
[29] Jeulin, T., (Semi-martingales et Grossissement d’une Filtration. Semi-martingales et Grossissement d’une Filtration, Lecture Notes in Mathematics, vol. 833 (1980), Springer: Springer Berlin), MR MR604176 (82h:60106) · Zbl 0444.60002
[30] Jeulin, T.; Yor, M., Grossissement d’une filtration et semi-martingales: formules explicites, (Séminaire de Probabilités, XII (Univ. Strasbourg, Strasbourg, 1976/1977). Séminaire de Probabilités, XII (Univ. Strasbourg, Strasbourg, 1976/1977), Lecture Notes in Math., vol. 649 (1978), Springer: Springer Berlin), 78-97, MR MR519998 · Zbl 0411.60045
[31] Jeulin, T.; Yor, M., Inégalité de Hardy, semimartingales, et faux-amis, (Séminaire de Probabilités de Strasbourg, Vol. 13 (1979)), 332-359, (fre) · Zbl 0419.60049
[32] Kantorovich, L. V., On the transfer of masses, Dokl. Akad. Nauk. SSSR, 37, 7-8, 227-229 (1942)
[33] Karatzas, I.; Lehoczky, J. P.; Shreve, S. E., Optimal portfolio and consumption decisions for a “small investor” on a finite horizon, SIAM J. Control Optim., 25, 6, 1557-1586 (1987), MR 912456 · Zbl 0644.93066
[34] Karatzas, I.; Lehoczky, J. P.; Shreve, S. E.; Xu, G.-L., Martingale and duality methods for utility maximization in an incomplete market, SIAM J. Control Optim., 29, 3, 702-730 (1991), MR 1089152 · Zbl 0733.93085
[35] Kchia, Y.; Protter, P., Progressive filtration expansions via a process, with applications to insider trading, Int. J. Theor. Appl. Finance, 18, 4, 1550027 (2015), 48, MR 3358108 · Zbl 1344.60037
[36] Kellerer, H. G., Duality theorems for marginal problems, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 67, 4, 399-432 (1984), MR 761565 · Zbl 0535.60002
[37] Kurtz, T. G., Weak and strong solutions of general stochastic models, Electron. Commun. Probab., 19, 58, 16 (2014), MR 3254737 · Zbl 1301.60035
[38] Lamberton, D.; Pagès, G., Sur l’approximation des réduites, Ann. Inst. Henri Poincaré Probab. Stat., 26, 2, 331-355 (1990), MR 1063754 · Zbl 0704.60042
[39] R. Lassalle, Causal transference plans and their Monge-Kantorovich problems, Submitted, arXiv:1303.6925.v2, 2015.
[40] Léonard, C., Girsanov theory under a finite entropy condition, (Séminaire de Probabilités XLIV. Séminaire de Probabilités XLIV, Lecture Notes in Math., vol. 2046 (2012), Springer: Springer Heidelberg), 429-465, MR 2953359 · Zbl 1253.60051
[41] Mansuy, R.; Yor, M., Random Times and Enlargements of Filtrations in a Brownian Setting (2006), Springer · Zbl 1103.60003
[42] Monge, G., Mémoire sur la Théorie des Déblais et des Remblais (1781), Histoire de l’Académie royale des sciences de Paris
[43] Peyré, G.; Cuturi, M., Computational optimal transport, Found. Trends Mach. Learn., 11, 5-6, 355-607 (2019)
[44] Pflug, G. Ch., Version-independence and nested distributions in multistage stochastic optimization, SIAM J. Optim., 20, 3, 1406-1420 (2009) · Zbl 1198.90307
[45] Pflug, G. Ch.; Pichler, A., A distance for multistage stochastic optimization models, SIAM J. Optim., 22, 1, 1-23 (2012), MR 2902682 · Zbl 1262.90118
[46] Pflug, G. Ch.; Pichler, A., (Multistage Stochastic Optimization. Multistage Stochastic Optimization, Springer Series in Operations Research and Financial Engineering (2014), Springer: Springer Cham), MR 3288310 · Zbl 1317.90220
[47] Pikovsky, I.; Karatzas, I., Anticipative portfolio optimization, Adv. Appl. Probab., 28, 4, 1095-1122 (1996), MR 1418248 · Zbl 0867.90013
[48] Protter, P., (Stochastic Integration and Differential Equations. Stochastic Integration and Differential Equations, Applications of Mathematics (New York) (2004), Springer-Verlag: Springer-Verlag Berlin), MR MR1037262 (91i:60148) · Zbl 1041.60005
[49] Rao, M.; Ren, Z., (Theory of Orlicz Spaces. Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146 (1991), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York), MR 1113700 · Zbl 0724.46032
[50] Sion, M., On general minimax theorems, Pacific J. Math., 8, 171-176 (1958), MR 0097026 · Zbl 0081.11502
[51] Villani, C., Topics in Optimal Transportation, no. 58 (2003), American Mathematical Soc. · Zbl 1106.90001
[52] Xu, G.-L.; Shreve, S. E., A duality method for optimal consumption and investment under short-selling prohibition. I. General market coefficients, Ann. Appl. Probab., 2, 1, 87-112 (1992), MR 1143394 · Zbl 0745.93083
[53] Yamada, T.; Watanabe, S., On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., 11, 1, 155-167 (1971) · Zbl 0236.60037
[54] Yor, M., Entropie d’une partition, et grossissement initial d’une filtration, (Grossissements de Filtrations: Exemples et Applications (1985), Springer), 45-58 · Zbl 0568.60050
[55] Yor, M., (Some Aspects of Brownian Motion. Part II. Some Aspects of Brownian Motion. Part II, Lectures in Mathematics ETH Zürich (1997), Birkhäuser Verlag: Birkhäuser Verlag Basel), Some recent martingale problems. MR 1442263 · Zbl 0880.60082
[56] Zaev, D., On the Monge-Kantorovich problem with additional linear constraints, Mat. Zametki, 98, 5, 664-683 (2015), MR 3438523 · Zbl 1336.49056
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.