# Variational approach to rare event simulation using least-squares regression

## TL;DR

This paper introduces an adaptive importance sampling method for rare event simulation in diffusion processes, leveraging a variational principle and stochastic control to optimize the change of measure.

## Contribution

It develops a novel variational framework using Gibbs principles to determine optimal importance sampling measures via stochastic control and approximation.

## Key findings

- Effective in high-dimensional settings
- Reduces variance in rare event estimates
- Demonstrated on toy examples

## Abstract

We propose an adaptive importance sampling scheme for the simulation of rare events when the underlying dynamics is given by a diffusion. The scheme is based on a Gibbs variational principle that is used to determine the optimal (i.e. zero-variance) change of measure and exploits the fact that the latter can be rephrased as a stochastic optimal control problem. The control problem can be solved by a stochastic approximation algorithm, using the Feynman-Kac representation of the associated dynamic programming equations, and we discuss numerical aspects for high-dimensional problems along with simple toy examples.