  In this paper, we study a semi-martingale optimal transport problem and its
application to the calibration of Local-Stochastic Volatility (LSV) models.
Rather than considering the classical constraints on marginal distributions at
initial and final time, we optimise our cost function given the prices of a
finite number of European options. We formulate the problem as a convex
optimisation problem, for which we provide a PDE formulation along with its
dual counterpart. Then we solve numerically the dual problem, which involves a
fully non-linear Hamilton-Jacobi-Bellman equation. The method is tested by
calibrating a Heston-like LSV model with simulated data and foreign exchange
market data.
