# Rare Event Simulation for Steady-State Probabilities via Recurrency Cycles

## TL;DR

This paper introduces Recurrent Multilevel Splitting (RMS), a novel algorithm leveraging the recurrent structure of Markov chains to efficiently estimate rare event probabilities in steady-state systems, significantly outperforming traditional Monte Carlo methods.

## Contribution

The paper presents RMS, a new algorithm that combines recurrence properties with multilevel splitting to improve rare event probability estimation in continuous-state Markov processes.

## Key findings

- RMS achieves several orders of magnitude efficiency gains over Monte Carlo.
- Numerical experiments validate RMS's effectiveness on complex stochastic models.
- The method is applicable to systems with nonlinear dynamics and climate-like characteristics.

## Abstract

We develop a new algorithm for the estimation of rare event probabilities associated with the steady-state of a Markov stochastic process with continuous state space $\mathbb R^d$ and discrete time steps (i.e. a discrete-time $\mathbb R^d$-valued Markov chain). The algorithm, which we coin Recurrent Multilevel Splitting (RMS), relies on the Markov chain's underlying recurrent structure, in combination with the Multilevel Splitting method. Extensive simulation experiments are performed, including experiments with a nonlinear stochastic model that has some characteristics of complex climate models. The numerical experiments show that RMS can boost the computational efficiency by several orders of magnitude compared to the Monte Carlo method.