de Finetti Risk Seminar

W. Schachermayer

MARTINGALE TRANSPORT, DE MARCH–TOUZI PAVING, AND STRETCHED BROWNIAN MOTION IN ℝ^d

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 ℝ^d, 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 ℝ^d.

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.