Variance Comparison between Infinitesimal Perturbation Analysis and Likelihood Ratio Estimators to Stochastic Gradient

We theoretically compare variances between the Infinitesimal Perturbation Analysis (IPA) estimator and the Likelihood Ratio (LR) estimator to Monte Carlo gradient for stochastic systems. The conditions proposed in [Cui et al., 2020] when the IPA estimator has a smaller variance can yield sharper inequalities or be further relaxed. We also prove a practically interesting result that the IPA estimators to European vanilla and arithmetic Asian options' Delta, respectively, have smaller variance when the underlying asset's return process is independent of the initial price.