We study counterfactual gradient estimation of conditional loss functionals of diffusion processes. In quantitative finance, these gradients are known as conditional Greeks: the sensitivity of expected market values, conditioned on some event of interest. The difficulty is that when the conditioning event has vanishing or zero probability, naive Monte Carlo estimators are prohibitively inefficient; kernel smoothing, though common, suffers from slow convergence. We propose a two-stage kernel-free methodology. First, we show using Malliavin calculus that the conditional loss functional of a diffusion process admits an exact representation as a Skorohod integral, yielding classical Monte-Carlo estimator variance and convergence rates. Second, we establish that a weak derivative estimate of the conditional loss functional with respect to model parameters can be evaluated algorithmically with constant variance, in contrast to the widely used score function method whose variance grows linearly in the sample path length. Together, these results yield an efficient framework for counterfactual conditional stochastic gradient algorithms and financial Greek computations in rare-event regimes.

翻译：本研究探讨扩散过程条件损失泛函的反事实梯度估计问题。在量化金融领域，此类梯度被称为条件希腊值：即在特定关注事件条件下，期望市场价值的敏感性。研究难点在于，当条件事件具有消失概率或零概率时，朴素蒙特卡洛估计器将面临难以接受的低效性问题；而常用的核平滑方法则存在收敛速度缓慢的缺陷。本文提出一种两阶段无核方法：首先，我们运用Malliavin分析证明扩散过程的条件损失泛函可精确表示为Skorohod积分，从而获得经典蒙特卡洛估计器的方差与收敛速率；其次，我们论证了条件损失泛函关于模型参数的弱导数估计可通过算法实现恒定方差计算，这与广泛使用的得分函数方法形成鲜明对比——后者的方差随样本路径长度线性增长。综合这些成果，本研究为罕见事件机制下的反事实条件随机梯度算法及金融希腊值计算构建了高效框架。