de Finetti Risk Seminar
May 5, 2022
May 5, 2022
W. Schachermayer
MARTINGALE TRANSPORT, DE MARCH–TOUZI PAVING, AND STRETCHED BROWNIAN MOTION IN ℝd
In classical optimal transport, the contributions of Benamou–Brenier and Mc-Cann regarding the time-dependent version of the problem are cornerstones of the field and form the basis for a variety of applications in other mathematical areas.
For µ,ν probability measures on ℝd, increasing in convex order, stretched Brownian motion [1] provides an analogue for the martingale version of this problem.
In dimension d=1 it was shown in [2] that any martingale transport decomposes into at most countably many invariant intervals and that this decomposition is universal. Extensions of this result to d≥2 were studied in [4], [5], [3].
We show that the dual optimization problem attached to a stretched Brownian motion induces the universal DeMarch–Touzi paving [3] of ℝd.
Joint work with M. Beiglböck, J. Backhoff, and B. Tschiderer.
REFERENCES
[1] J. Backhoff-Veraguas, M. Beiglböck, M. Huesmann, and S. Källblad. Martingale Benamou-Brenier: a probabilistic perspective. Ann. Probab., 48(5):2258–2289, 2020.
[2] M. Beiglböck and N. Juillet. On a problem of optimal transport under marginal martingale constraints. Ann. Probab., 44(1):42–106, 2016.
[3] H. De March and N. Touzi. Irreducible convex paving for decomposition of multidimensional martingale transport plans. Ann. Probab., 47(3):1726–1774, 2019.
[4] N. Ghoussoub, Y.-H. Kim, and T. Lim. Structure of optimal martingale transport plans in general dimensions. Ann. Probab., 47(1):109–164, 2019.
[5] J. Obłój and P. Siorpaes. Structure of martingale transports in finite dimensions. arXiv:1702.08433, 2017.