Stochastic Derivative Estimation for Discontinuous Sample Performances: A Leibniz Integration Perspective

TL;DR

This paper introduces a new stochastic derivative estimation method for discontinuous sample performance functions using Leibniz integral rule, enabling unbiased and single-run estimators by embedding indicator functions and handling domain boundaries.

Contribution

It generalizes the GLR method by incorporating boundary surface integrals and provides conditions for their vanishing, improving derivative estimation for discontinuous functions.

Findings

Single-run unbiased derivative estimator for discontinuous functions.

The method generalizes the GLR approach with boundary surface integral considerations.

Numerical experiments confirm the robustness and effectiveness of the proposed estimators.

Abstract

We develop a novel stochastic derivative estimation framework for sample performance functions that are discontinuous in the parameter of interest, based on the multidimensional Leibniz integral rule. When discontinuities arise from indicator functions, we embed the indicator functions into the sample space, yielding a continuous performance function over a parameter-dependent domain. Applying the Leibniz integral rule in this case produces a single-run, unbiased derivative estimator. For general discontinuous functions, we apply a change of variables to shift parameter dependence into the sample space and the underlying probability measure. Applying the Leibniz integral rule leads to two terms: a standard likelihood ratio (LR) term from differentiating the underlying probability measure and a surface integral from differentiating the boundary of the domain. Evaluating the surface integral may require simulating multiple sample paths. Our proposed Leibniz integration framework generalizes the generalized LR (GLR) method and provides intuition as to when the surface integral vanishes, thereby enabling single-run, easily implementable estimators. Numerical experiments demonstrate the effectiveness and robustness of our methods.