  We solve constrained optimal transport problems between the laws of solutions
of stochastic differential equations (SDEs). We consider SDEs with irregular
coefficients, making only minimal regularity assumptions. We show that the
so-called synchronous coupling is optimal among bicausal couplings, that is
couplings that respect the flow of information encoded in the stochastic
processes. Our results provide a method to numerically compute the adapted
Wasserstein distance between laws of SDEs with irregular coefficients.
Moreover, we introduce a transformation-based semi-implicit numerical scheme
and establish the first strong convergence result for SDEs with exponentially
growing and discontinuous drift.
