We propose a new algorithm for variance reduction when estimating $f(X_T)$ where $X$ is the solution to some stochastic differential equation and $f$ is a test function. The new estimator is $(f(X^1_T) + f(X^2_T))/2$, where $X^1$ and $X^2$ have same marginal law as $X$ but are pathwise correlated so that to reduce the variance. The optimal correlation function $\rho$ is approximated by a deep neural network and is calibrated along the trajectories of $(X^1, X^2)$ by policy gradient and reinforcement learning techniques. Finding an optimal coupling given marginal laws has links with maximum optimal transport.

翻译：我们提出一种新的方差缩减算法，用于估计$f(X_T)$，其中$X$为随机微分方程的解，$f$为测试函数。该新估计量为$(f(X^1_T) + f(X^2_T))/2$，其中$X^1$与$X^2$具有与$X$相同的边际分布，但通过路径相关性设计实现方差缩减。最优相关函数$\rho$通过深度神经网络近似，并沿$(X^1, X^2)$轨迹采用策略梯度与强化学习技术进行校准。在给定边际分布条件下寻找最优耦合与最大最优输运问题存在内在联系。