# Splitting Algorithms for Rare Events of Semimartingale Reflecting Brownian motions

## TL;DR

This paper develops a large deviation framework and particle-based simulation algorithms for estimating rare event probabilities in high-dimensional semimartingale reflecting Brownian motions, demonstrating stability and superiority over standard Monte Carlo methods.

## Contribution

It introduces a novel large deviation approach and constructs subsolutions for rare event simulation in SRBMs, improving estimation efficiency and stability.

## Key findings

- The probability of the rare event satisfies a large deviation principle.
- The proposed particle-based algorithm is stable and theoretically superior to standard Monte Carlo.
- The method applies broadly to positive recurrent SRBMs.

## Abstract

We study rare event simulations of semimartingale reflecting Brownian motions (SRBMs) in an orthant. The rare event of interest is that a $d$-dimensional positive recurrent SRBM enters the set $B_n = \{z\in\mathbb{R}^d: \sum_{k=1}^d z_k = n\}$ before reaching a small neighborhood of the origin as $n\to\infty$. We show that under a proper scaling and some regularity conditions, the probability of interest satisfies a large deviation principle. We then construct a subsolution to the variational problem for our rare event, and based on this subsolution construct particle based simulation algorithms to estimate the probability of the rare event. It is shown that the proposed algorithm is stable and theoretically superior to standard Monte Carlo for a broad class of positive recurrent SRBMs.