  We develop a non-parametric, optimal transport driven, calibration
methodology for local volatility models with stochastic interest rate. The
method finds a fully calibrated model which is the closest to a given reference
model. We establish a general duality result which allows to solve the problem
via optimising over solutions to a non-linear HJB equation. We then apply the
method to a sequential calibration setup: we assume that an interest rate model
is given and is calibrated to the observed term structure in the market. We
then seek to calibrate a stock price local volatility model with volatility
coefficient depending on time, the underlying and the short rate process, and
driven by a Brownian motion which can be correlated with the randomness driving
the rates process. The local volatility model is calibrated to a finite number
of European options prices via a convex optimisation problem derived from the
PDE formulation of semimartingale optimal transport. Our methodology is
analogous to Guo, Loeper, and Wang, 2022 and Guo, Loeper, Obloj, et al., 2022a
but features a novel element of solving for discounted densities, or
sub-probability measures. We present numerical experiments and test the
effectiveness of the proposed methodology.
