Specific Wasserstein Divergence Between Continuous Martingales

References

• [1] Acciaio B, Backhoff-Veraguas J, Zalashko A (2020) Causal optimal transport and its links to enlargement of filtrations and continuous-time stochastic optimization. Stochastic Processes Appl. 130(5):2918–2953.Crossref, Google Scholar
• [2] Aldous D (2023) What is the max-entropy win-probability martingale? Accessed May 2023, https://www.stat.berkeley.edu/∼aldous/Research/OP/max_ent_mg.html.Google Scholar
• [3] Aldous DJ (2013) Using prediction market data to illustrate undergraduate probability. Amer. Math. Monthly 120(7):583–593.Crossref, Google Scholar
• [4] Avellaneda M, Friedman C, Holmes R, Samperi D (2001) Calibrating volatility surfaces via relative-entropy minimization. Quant. Finance 1(1):42–51.Google Scholar
• [5] Backhoff J, Wang Z, Zhang X (2024) Exciting games and Monge-Ampère equations. Preprint, submitted December 2, https://arxiv.org/abs/2412.01995v1.Google Scholar
• [6] Backhoff J, Beiglböck M, Lin Y, Zalashko A (2017) Causal transport in discrete time and applications. SIAM J. Optim. 27(4):2528–2562.Crossref, Google Scholar
• [7] Backhoff J, Conforti G, Gentil I, Léonard C (2020) The mean field Schrödinger problem: Ergodic behavior, entropy estimates and functional inequalities. Probab. Theory Related Fields 178(1):475–530.Crossref, Google Scholar
• [8] Backhoff-Veraguas J, Beiglböck M (2024) The most exciting game. Electronic Commun. Probab. 29:1–12.Crossref, Google Scholar
• [9] Backhoff-Veraguas J, Belloto E (2024) Multidimensional specific relative entropy between continuous martingales. Preprint, submitted November 18, https://arxiv.org/abs/2411.11408.Google Scholar
• [10] Backhoff-Veraguas J, Pammer G (2022) Applications of weak transport theory. Bernoulli 28(1):370–394.Crossref, Google Scholar
• [11] Backhoff-Veraguas J, Unterberger C (2023) On the specific relative entropy between martingale diffusions on the line. Electronic Comm. Probab. 28:1–12.Crossref, Google Scholar
• [12] Backhoff-Veraguas J, Källblad S, Robinson BA (2025) Adapted Wasserstein distance between the laws of SDEs. Stochastic Processes Appl. 189:104689.Crossref, Google Scholar
• [13] Backhoff-Veraguas J, Lacker D, Tangpi L (2020) Nonexponential Sanov and Schilder theorems on Wiener space: BSDEs, Schrödinger problems and control. Ann. Appl. Probab. 30(3):1321–1367.Crossref, Google Scholar
• [14] Backhoff-Veraguas J, Bartl D, Mathias B, Manu E (2020) Adapted Wasserstein distances and stability in mathematical finance. Finance Stochastics 24(3):601–632.Crossref, Google Scholar
• [15] Backhoff-Veraguas J, Beiglböck M, Huesmann M, Källblad S (2020) Martingale Benamou–Brenier. Ann. Probab. 48(5):2258–2289.Crossref, Google Scholar
• [16] Backhoff-Veraguas J, Beiglböck M, Schachermayer W, Tschiderer B (2026) Existence of Bass martingales and the martingale Benamou Brenier problem in Rd. Ann. Probab. Forthcoming.Google Scholar
• [17] Bartl D, Wiesel J (2023) Sensitivity of multiperiod optimization problems with respect to the adapted Wasserstein distance. SIAM J. Financial Math. 14(2):704–720.Crossref, Google Scholar
• [18] Bartl D, Beiglböck M, Pammer G (2024) The Wasserstein space of stochastic processes. J. Eur. Math. Soc., ePub ahead of print December 16, https://dx.doi.org/10.4171/JEMS/1554.Crossref, Google Scholar
• [19] Beiglböck M, Juillet N (2016) On a problem of optimal transport under marginal martingale constraints. Ann. Probab. 44(1):42–106.Crossref, Google Scholar
• [20] Beiglböck M, Juillet N (2021) Shadow couplings. Trans. Amer. Math. Soc. 374(7):4973–5002.Crossref, Google Scholar
• [21] Beiglböck M, Cox A, Huesmann M (2017) Optimal transport and Skorokhod embedding. Inventions Math. 208(2):327–400.Crossref, Google Scholar
• [22] Beiglböck M, Henry-Labordère P, Penkner F (2013) Model-independent bounds for option prices: A mass transport approach. Finance Stochastics 17(3):477–501.Crossref, Google Scholar
• [23] Beiglböck M, Nutz M, Stebegg F (2022) Fine properties of the optimal Skorokhod embedding problem. J. Eur. Math. Soc. 24(4):1389–1429.Crossref, Google Scholar
• [24] Beiglböck M, Nutz M, Touzi N (2017) Complete duality for martingale optimal transport on the line. Ann. Probab. 45(5):3038–3074.Crossref, Google Scholar
• [25] Benamou JD, Chazareix G, Loeper G (2024) From entropic transport to martingale transport, and applications to model calibration. Preprint, submitted June 17, https://arxiv.org/abs/2406.11537.Google Scholar
• [26] Benamou JD, Carlier G, Di Marino S, Nenna L (2019) An entropy minimization approach to second-order variational mean-field games. Math. Models Methods Appl. Sci. 29(8):1553–1583.Crossref, Google Scholar
• [27] Benamou JD, Chazareix G, Hoffmann M, Loeper G, Vialard FX (2024) Entropic semi-martingale optimal transport. Preprint, submitted December 16, https://arxiv.org/abs/2408.09361.Google Scholar
• [28] Bouchard B, Nutz M (2015) Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab. 25(2):823–859.Crossref, Google Scholar
• [29] Chen Y, Georgiou TT, Pavon M (2016) On the relation between optimal transport and Schrödinger bridges: A stochastic control viewpoint. J. Optim. Theory Appl. 169(2):671–691.Crossref, Google Scholar
• [30] Cohen A, Dolinsky Y (2022) A scaling limit for utility indifference prices in the discretised Bachelier model. Finance Stochastics 26(2):335–358.Crossref, Google Scholar
• [31] Conforti G (2019) A second order equation for Schrödinger bridges with applications to the hot gas experiment and entropic transportation cost. Probab. Theory Related Fields 174(1–2):1–47.Crossref, Google Scholar
• [32] Dolinsky Y, Soner HM (2014) Martingale optimal transport and robust hedging in continuous time. Probab. Theory Related Fields 160(1–2):391–427.Crossref, Google Scholar
• [33] Dolinksy Y, Zhang X (2025) Scaling limits for exponential hedging in the Brownian Framework. Preprint, submitted February 24, https://arxiv.org/abs/2502.17186v1.Google Scholar
• [34] Eckstein S (2019) Extended Laplace principle for empirical measures of a Markov chain. Adv. Appl. Probab. 51(1):136–167.Crossref, Google Scholar
• [35] Ely J (2017) Beeps. Amer. Econom. Rev. 107(1):31–53.Crossref, Google Scholar
• [36] Ely J, Frankel A, Kamenica E (2015) Suspense and surprise. J. Polit. Econom. 123(1):215–260.Crossref, Google Scholar
• [37] Figalli A, Glaudo F (2021) An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows, EMS Textbooks in Mathematics (EMS Press, Berlin).Crossref, Google Scholar
• [38] Föllmer H (2022) Optimal couplings on Wiener space and an extension of Talagrand’s transport inequality. Yin G, Zariphopoulou T, eds. Stochastic Analysis, Filtering, and Stochastic Optimization (Springer, Cham, Switzerland), 147–175.Crossref, Google Scholar
• [39] Föllmer H, Penner I (2006) Convex risk measures and the dynamics of their penalty functions. Statist. Risk Modeling 24(1):61–96.Crossref, Google Scholar
• [40] Galichon A, Henry-Labordère P, Touzi N (2014) A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options. Ann. Appl. Probab. 24(1):312–336.Crossref, Google Scholar
• [41] Gantert N (1991) Einige Grosse Abweichungen der Brownschen Bewegung, Bonner Mathematische Schriften (Mathematischen Institut der Universität, Bonn, German).Google Scholar
• [42] Gozlan N, Léonard C (2010) Transport inequalities. A survey. Markov Processes Related Fields 16(4):635–736.Google Scholar
• [43] Guo I, Loeper G (2021) Path dependent optimal transport and model calibration on exotic derivatives. Ann. Appl. Probab. 31(3):1232–1263.Crossref, Google Scholar
• [44] Guo G, Howison SD, Possamaï D, Reisinger C (2025) Randomness and early termination: What makes a game exciting? Probab. Theory Related Fields 192(1):135–162.Crossref, Google Scholar
• [45] Henry-Labordere P (2019) From (martingale) Schrödinger bridges to a new class of stochastic volatility model. Preprint, submitted March 15, https://dx.doi.org/10.2139/ssrn.3353270.Google Scholar
• [46] Hobson DG (1998) Volatility misspecification, option pricing and superreplication via coupling. Ann. Appl. Probab. 8(1):193–205.Crossref, Google Scholar
• [47] Hobson D, Neuberger A (2012) Robust bounds for forward start options. Math. Finance 22(1):31–56.Crossref, Google Scholar
• [48] Huesmann M, Trevisan D (2019) A Benamou–Brenier formulation of martingale optimal transport. Bernoulli 25(4A):2729–2757.Crossref, Google Scholar
• [49] Kallenberg O (2002) Foundations of Modern Probability, Probability and Its Applications, 2nd ed. (Springer-Verlag, New York).Crossref, Google Scholar
• [50] Karandikar RL (1995) On pathwise stochastic integration. Stochastic Processes Appl. 57(1):11–18.Crossref, Google Scholar
• [51] Karatzas I, Shreve SE (1991) Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, 2nd ed., vol. 113 (Springer-Verlag, New York).Google Scholar
• [52] Karlin S, Taylor HM (1981) A Second Course in Stochastic Processes (Academic Press, New York).Google Scholar
• [53] Lacker D (2020) A non-exponential extension of Sanov’s theorem via convex duality. Adv. Appl. Probab. 52(1):61–101.Crossref, Google Scholar
• [54] Lassalle R (2018) Causal transport plans and their Monge–Kantorovich problems. Stochastic Anal. Appl. 36(3):452–484.Crossref, Google Scholar
• [55] Léonard C (2014) A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete Continuous Dynam. Systems 34(4):1533–1574.Crossref, Google Scholar
• [56] Léonard C, Rœlly S, Zambrini JC (2014) Reciprocal processes. A measure-theoretical point of view. Probab. Surveys 11:237–269.Crossref, Google Scholar
• [57] Loeper G (2018) Option pricing with linear market impact and nonlinear Black–Scholes equations. Ann. Appl. Probab. 28(5):2664–2726.Crossref, Google Scholar
• [58] Montero M (2023) Merge of two oppositely biased Wiener processes. Preprint, submitted March 31, https://dx.doi.org/10.48550/arXiv.2303.18088.Google Scholar
• [59] Nutz M, Wiesel J (2026) On the martingale Schrödinger bridge between two distributions. Bernoulli. Forthcoming.Google Scholar
• [60] Nutz M, Wiesel J, Zhao L (2023) Martingale Schrödinger bridges and optimal semistatic portfolios. Finance Stochastics 27(1):233–254.Crossref, Google Scholar
• [61] Pal S, Wong TKL (2020) Multiplicative Schrödinger problem and the Dirichlet transport. Probab. Theory Related Fields 178(1–2):613–654.Crossref, Google Scholar
• [62] Pflug G (2010) Version-independence and nested distributions in multistage stochastic optimization. SIAM J. Optim. 20(3):1406–1420.Crossref, Google Scholar
• [63] Pflug G, Pichler A (2012) A distance for multistage stochastic optimization models. SIAM J. Optim. 22(1):1–23.Crossref, Google Scholar
• [64] Santambrogio F (2015) Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling, Progress in Nonlinear Differential Equations and Their Applications, vol. 87 (Birkhäuser, Cham, Switzerland).Crossref, Google Scholar
• [65] Schrödinger E (1931) Über die Umkehrung der Naturgesetze, Sitzungsberichte der Preussischen Akademie der Wissenschaften. Physikalisch-mathematische Klasse.Google Scholar
• [66] Schrödinger E (1932) Sur la théorie relativiste de l’électron et l’interprétation de la mécanique quantique. Ann. Inst. H. Poincaré 2(4):269–310.Google Scholar
• [67] Tan X, Touzi N (2013) Optimal transportation under controlled stochastic dynamics. Ann. Probab. 41(5):3201–3240.Crossref, Google Scholar
• [68] Vázquez JL (2007) The Porous Medium Equation: Mathematical Theory, Oxford Mathematical Monographs (The Clarendon Press, Oxford University Press, Oxford, UK).Google Scholar
• [69] Villani C (2003) Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58 (American Mathematical Society, Providence, RI).Crossref, Google Scholar
• [70] Villani C (2008) Optimal Transport: Old and New, vol. 338 (Springer Science & Business Media, Berlin).Google Scholar
• [71] Zhong W (2022) Optimal dynamic information acquisition. Econometrica 90(4):1537–1582.Crossref, Google Scholar