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

Xingyu Ren, Michael C. Fu, Pierre L'Ecuyer

TL;DR

This work develops a stochastic derivative estimation framework for discontinuous sample performances by exploiting the multidimensional Leibniz integral rule. For discontinuities from indicators, it embeds the indicator into the sample space to produce a θ-dependent domain and derives a single-run unbiased estimator via the Leibniz divergence form; for general discontinuities, it uses a push-out change of variables to separate the θ-dependence into the density, yielding a volume (LR) term plus a surface (boundary) term that may require multiple sample paths. The framework generalizes GLR, offers weaker verifiability conditions, and provides geometric intuition on when the surface term vanishes, enabling simpler estimators. It is supported by simulation evidence showing robustness of the Leibniz divergence estimator under dependence, and guidance on when to avoid surface terms through hyperrectangle domains or suitable transformations. Overall, the paper broadens the toolkit for stochastic sensitivity analysis under discontinuities, with practical implications for simulation optimization and risk management.

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.

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

TL;DR

This work develops a stochastic derivative estimation framework for discontinuous sample performances by exploiting the multidimensional Leibniz integral rule. For discontinuities from indicators, it embeds the indicator into the sample space to produce a θ-dependent domain and derives a single-run unbiased estimator via the Leibniz divergence form; for general discontinuities, it uses a push-out change of variables to separate the θ-dependence into the density, yielding a volume (LR) term plus a surface (boundary) term that may require multiple sample paths. The framework generalizes GLR, offers weaker verifiability conditions, and provides geometric intuition on when the surface term vanishes, enabling simpler estimators. It is supported by simulation evidence showing robustness of the Leibniz divergence estimator under dependence, and guidance on when to avoid surface terms through hyperrectangle domains or suitable transformations. Overall, the paper broadens the toolkit for stochastic sensitivity analysis under discontinuities, with practical implications for simulation optimization and risk management.

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.

Paper Structure

This paper contains 22 sections, 11 theorems, 88 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Leibniz Rules in ${\mathbb{R}}^n$ with Motivating Examples
  3. Discontinuous Sample Performances due to Indicator Functions
  4. Combining the Leibniz Integral Rule with the Push-out LR Method
  5. Extensions of the Leibniz Integral Estimator
  6. Hyperrectangle Support $\Omega$
  7. Local Change of Variables
  8. Easier-to-Verify Alternative Conditions
  9. Unbounded Sample Space $\Omega$
  10. Almost Everywhere Differentiable $\varphi$
  11. Simulation Example
  12. Conclusions
  13. A Geometric Interpretation of the Leibniz Integral Rule
  14. Copulas and Derivations of Estimators in Examples
  15. Copulas
  16. ...and 7 more sections

Key Result

Lemma 1

Let $D_\theta\subset{\mathbb{R}}^n$ be a bounded set, and let $U\subset {\mathbb{R}}^n$ be a $\theta$-independent bounded set. Suppose there exists a function $\phi:U\times\Theta\mapsto{\mathbb{R}}^n$, continuously differentiable in both arguments, such that $D_\theta=\phi(U,\theta)$. Assume that fo where $\vec{v}(x,\theta) = \partial_\theta \phi(u,\theta)|_{u=\phi^{-1}(x,\theta)}$, $\vec{n}(x,\th

Figures (1)

  • Figure EC.1: The original domain $D_\theta$ and the perturbed domain $D_{\theta+\Delta\theta}$.

Theorems & Definitions (25)

  • Lemma 1
  • Remark 1
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Theorem 1
  • Remark 2
  • Example 5
  • Remark 3
  • ...and 15 more