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Improved Convergence Rate of Nested Simulation with LSE on Sieve

Ruoxue Liu, Liang Ding, Wenjia Wang, Lu Zou

TL;DR

This work extends kernel-based nested simulation by reframing conditional expectation estimation as Least Squares Estimation on a sieve, unifying KRR with inducing points and neural networks under a common nonparametric framework. The authors derive high-probability and $L^p$ convergence results for general functionals $\mathcal{T}$, including nested expectations and risk measures like VaR, with rates governed by the sieve complexity and smoothness assumptions. Through theoretical results and numerical experiments, they show that LSE on sieve can achieve the optimal $\mathcal{O}(\Gamma^{-1/2})$ convergence under mild conditions, often outperforming standard Monte Carlo and KRR, while offering favorable computational costs via inducing points. The findings provide a flexible toolkit for efficient nested simulation across applications such as risk measurement and portfolio optimization, with clear guidance on when to prefer KRR-inducing-point or neural-network sieves and how smoothness and topology influence rates.

Abstract

Nested simulation encompasses the estimation of functionals linked to conditional expectations through simulation techniques. In this paper, we treat conditional expectation as a function of the multidimensional conditioning variable and provide asymptotic analyses of general Least Squared Estimators on sieve, without imposing specific assumptions on the function's form. Our study explores scenarios in which the convergence rate surpasses that of the standard Monte Carlo method and the one recently proposed based on kernel ridge regression. We also delve into the conditions that allow for achieving the best possible square root convergence rate among all methods. Numerical experiments are conducted to support our statements.

Improved Convergence Rate of Nested Simulation with LSE on Sieve

TL;DR

This work extends kernel-based nested simulation by reframing conditional expectation estimation as Least Squares Estimation on a sieve, unifying KRR with inducing points and neural networks under a common nonparametric framework. The authors derive high-probability and convergence results for general functionals , including nested expectations and risk measures like VaR, with rates governed by the sieve complexity and smoothness assumptions. Through theoretical results and numerical experiments, they show that LSE on sieve can achieve the optimal convergence under mild conditions, often outperforming standard Monte Carlo and KRR, while offering favorable computational costs via inducing points. The findings provide a flexible toolkit for efficient nested simulation across applications such as risk measurement and portfolio optimization, with clear guidance on when to prefer KRR-inducing-point or neural-network sieves and how smoothness and topology influence rates.

Abstract

Nested simulation encompasses the estimation of functionals linked to conditional expectations through simulation techniques. In this paper, we treat conditional expectation as a function of the multidimensional conditioning variable and provide asymptotic analyses of general Least Squared Estimators on sieve, without imposing specific assumptions on the function's form. Our study explores scenarios in which the convergence rate surpasses that of the standard Monte Carlo method and the one recently proposed based on kernel ridge regression. We also delve into the conditions that allow for achieving the best possible square root convergence rate among all methods. Numerical experiments are conducted to support our statements.
Paper Structure (23 sections, 12 theorems, 99 equations, 2 figures, 3 tables)

This paper contains 23 sections, 12 theorems, 99 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Notation and Organization
  4. Problem Formulation
  5. LSE on Sieve
  6. Asymptotic Analysis
  7. Examples
  8. Kernel Ridge Regression with Inducing Points
  9. Nested Expectation
  10. Risk Measures
  11. Neural networks
  12. Numerical Experiments
  13. Synthetic Data
  14. Portfolio Risk Management
  15. Conclusion
  16. ...and 8 more sections

Key Result

Theorem 1

Suppose Assumptions assump:support-- assump:b_bounded hold. Take any upper bound $\Psi({\delta})\geqslant \mathcal{J}({\delta})$ in such a way that $\Psi({\delta})/{\delta}^2$ is a non-decreasing function for any $0<{\delta}<2^7{\sigma}/\sqrt{m}$. Then for a constant $C$ only depends on $\sigma$, an we have

Figures (2)

  • Figure 1: Illustration of Covering Sets. A $u$-covering of $\mathcal{M}$ is a collection of element $\{\theta_1, \cdots, \theta_N\}$, such that for each $\theta\in \mathcal{M}$, there exists a $j\in\{1,\cdots, N\}$, such that $\|\theta-\theta_j\|\leqslant \delta$. (Figure 5.1 in Wainwright19. )
  • Figure 2: Upper row: MSE error of experiments associated to $\mathcal{T}(Z)=Z^2$; middle: MSE errors of experiments associated to $\mathcal{T}(Z)=\text{VaR}(Z)$; bottom row: Computational time of experiments. The lines represent the averaged errors and the shaded area represents the STD for nested expectation and one-tenth of the STD for VaR.

Theorems & Definitions (29)

  • Definition 1
  • Remark 1
  • Remark 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 3
  • Remark 4
  • Proposition 1
  • Remark 5
  • ...and 19 more