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A Short Report on Importance Sampling for Rare Event Simulation in Diffusions

Zhiwei Gao

TL;DR

The paper addresses estimating rare-event functionals for diffusion processes in the small-noise regime by linking importance sampling to stochastic optimal control through the HJB equation. It derives a variational, log-transform framework where the optimal proposal minimizes variance, and proposes a practical cross-entropy method to approximate the optimal control via a parametric, basis-function representation of the value function. The approach yields a linear-in-parameters control and uses weighted CE updates to adapt the policy from trajectory data, with a 1D double-well example demonstrating substantial variance reduction. While showing log-efficiency in the limit, the work notes challenges in high dimensions, basis selection, and time-dependent controls, highlighting areas for future theoretical and computational development.

Abstract

In this manuscript, we investigate importance sampling methods for rare-event simulation in diffusion processes. We show, from a large-deviation perspective, that the resulting importance sampling estimator is log-efficient. This connection is established via a stochastic optimal control formulation, and the associated Hamilton--Jacobi--Bellman (HJB) equation is derived using dynamic programming. To approximate the optimal control, we adopt a spectral parameterization and employ the cross-entropy method to estimate the parameters by solving a least-squares problem. Finally, we present a numerical example to validate the effectiveness of the cross-entropy approach and the efficiency of the resulting importance sampling estimator.

A Short Report on Importance Sampling for Rare Event Simulation in Diffusions

TL;DR

The paper addresses estimating rare-event functionals for diffusion processes in the small-noise regime by linking importance sampling to stochastic optimal control through the HJB equation. It derives a variational, log-transform framework where the optimal proposal minimizes variance, and proposes a practical cross-entropy method to approximate the optimal control via a parametric, basis-function representation of the value function. The approach yields a linear-in-parameters control and uses weighted CE updates to adapt the policy from trajectory data, with a 1D double-well example demonstrating substantial variance reduction. While showing log-efficiency in the limit, the work notes challenges in high dimensions, basis selection, and time-dependent controls, highlighting areas for future theoretical and computational development.

Abstract

In this manuscript, we investigate importance sampling methods for rare-event simulation in diffusion processes. We show, from a large-deviation perspective, that the resulting importance sampling estimator is log-efficient. This connection is established via a stochastic optimal control formulation, and the associated Hamilton--Jacobi--Bellman (HJB) equation is derived using dynamic programming. To approximate the optimal control, we adopt a spectral parameterization and employ the cross-entropy method to estimate the parameters by solving a least-squares problem. Finally, we present a numerical example to validate the effectiveness of the cross-entropy approach and the efficiency of the resulting importance sampling estimator.

Paper Structure

This paper contains 7 sections, 1 theorem, 79 equations, 3 figures.

Table of Contents

  1. Problem setup
  2. Monte Carlo simulation
  3. Importance sampling
  4. Importance sampling and stochastic optimal control
  5. Cross entropy methods for importance sampling
  6. Numerical experiments
  7. Conclusion

Key Result

Lemma 4.1

dupuis2012importance Let $X^{\epsilon}$ be the solution of (2). Given $u\in \mathcal{U}$, the induced measure $\mathcal{Q}$ is given by Girsonov's theorem, i.e., and $B_{s} = W_{s} - \frac{1}{\sqrt{\epsilon}}\int_{t}^{s}u_{r}dr$ is a standard Brownian motion under $\mathbb{Q}$. Similarly, we introduce the control $-u$ and denote the induced measure as $\mathbb{Q}^{-u}$, where the new standard Bro

Figures (3)

  • Figure 1: The trajectories with uncontrolled SDE.
  • Figure 2: The trajectories for the SDE with learned control.
  • Figure 3: The original potential, modified potential and learned control.

Theorems & Definitions (3)

  • Definition 3.1
  • Lemma 4.1
  • proof