We solve constrained optimal transport problems in which the marginal laws are given by 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. We show that this can be applied to quantifying model uncertainty in stochastic optimisation problems. 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：我们解决了约束最优输运问题，其中边缘分布由随机微分方程（SDE）解的分布给出。我们考虑具有不规则系数的随机微分方程，仅施加最低限度的正则性假设。我们证明，所谓的同步耦合在双因果耦合中是最优的，即那些尊重随机过程所编码信息流的耦合。我们的结果为数值计算具有不规则系数的随机微分方程分布之间的适应Wasserstein距离提供了一种方法。我们表明，这可以应用于量化随机优化问题中的模型不确定性。此外，我们引入了一种基于变换的半隐式数值格式，并为具有指数增长且不连续漂移项的随机微分方程建立了首个强收敛结果。