Semimartingale optimal transport and applications

Abstract: In this talk, we introduce stochastic optimal transport and extend it to continuous time, path-dependent settings. The problem is to find a semimartingale measure that satisfies general path-dependent constraints, while minimising a cost function on the drift and diffusion coefficients. Duality is established via the Fenchel-Rockafellar duality theorem and expressed via non-linear path-dependent partial differential equations (PPDEs).
We then briefly discuss some applications in mathematical finance, including the calibration of path-dependent derivatives, LSV models, and joint SPX-VIX models. It produces a non-parametric volatility model that localises to the features of the derivatives. Another application is in the robust pricing and hedging of American options in continuous time, which is achieved by enriching the state space by the stopping decisions.