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.

翻译：我们解决了随机微分方程（SDEs）解律之间的约束最优输运问题。我们考虑了具有不规则系数的随机微分方程，仅需极少的正则性假设。我们证明，所谓的同步耦合在双因果耦合（即尊重随机过程中信息流的耦合）中是最优的。我们的结果为数值计算不规则系数随机微分方程解律之间的适应Wasserstein距离提供了方法。此外，我们提出了一种基于变换的半隐式数值格式，并首次建立了对于漂移项指数增长且不连续的随机微分方程的强收敛结果。