Abstract: The Schrodinger problem can be recast as an optimal transport problem where the diffusion coefficient is fixed and the control is the drift. It was originally stated as a relative entropy minimization under marginal constraints, and can therefore be solved by the celebrated Sinkhorn algorithm.

The Bass martingale problem on the other hand can be stated as an optimal transport problem where the diffusion coefficient is the control and the drift is set to zero (martingale contraint).

It was also recently found that it could be solved by a measure preserving version of the Sinkhorn algorithm, the MPMS algorithm introduced by Joseph, Loeper, Obloj in https://arxiv.org/abs/2310.13797.

It turns out that there is a continuum of optimal transport problems that link Bass and Schrodinger which we name the Schrodinger Bass Bridges (SBB). These problems lead to Sinkhorn like systems at optimality, and can also be solved efficiently even in high dimensions (e.g. for images).

This is based on joint works with A. Alouadi (UPC, BNP-PAR), P. Henry-Labordère (Qube RT), H. Pham (Ecole Polytechnique), O. Mazhar (UPC, LPSM)  and N. Touzi  (NYU), B. Joesph (genOTC), J. Obloj (University of Oxford).

Bio: Grégoire Loeper is a researcher in mathematics who has spent his career moving between academia and the financial industry. His early research focused on optimal transport and partial differential equations, where he addressed fundamental mathematical questions. He then moved to the financial industry at BNP Paribas and focused on practical problems in derivatives pricing and risk management, where he occupied various positions, from quantitative research to structuring and trading. He has contributed to developing rigorous calibration frameworks using semi-martingale optimal transport.

As Senior Scientific Advisor at BNP Paribas and founder of genOTC, having previously led academic programs at Monash University, he continues to explore how optimal transport connects mathematical theory with diverse practical applications, from financial markets to machine learning.