Variance reduction techniques are of crucial importance for the efficiency of Monte Carlo simulations in finance applications. We propose the use of neural SDEs, with control variates parameterized by neural networks, in order to learn approximately optimal control variates and hence reduce variance as trajectories of the SDEs are being simulated. We consider SDEs driven by Brownian motion and, more generally, by L\'{e}vy processes including those with infinite activity. For the latter case, we prove optimality conditions for the variance reduction. Several numerical examples from option pricing are presented.

翻译：方差缩减技术对于金融应用中蒙特卡洛模拟的效率至关重要。我们提出使用神经随机微分方程，并采用由神经网络参数化的控制变量，以学习近似最优的控制变量，从而在模拟随机微分方程轨迹时降低方差。我们考虑由布朗运动驱动的随机微分方程，以及更一般地由包括无限活动过程在内的莱维过程驱动的随机微分方程。对于后者，我们证明了方差缩减的最优性条件。文中还给出了来自期权定价的若干数值示例。