# Multilevel Monte Carlo theta EM scheme for SDDEs with small noise

## TL;DR

This paper introduces a multilevel Monte Carlo theta EM scheme tailored for stochastic differential delay equations with small noise, analyzing variance and moment properties under different Lipschitz conditions.

## Contribution

It develops a modified scheme under one-sided Lipschitz conditions to ensure moment finiteness, extending the applicability of the method.

## Key findings

- Variance of coupled paths derived under Lipschitz conditions
- Modified scheme guarantees moment finiteness
- Second moment estimates for coupled paths provided

## Abstract

In this paper, a multilevel Monte Carlo theta EM scheme is provided for stochastic differential delay equations with small noise. Under a global Lipschitz condition, the variance of two coupled paths is derived. Then, the global Lipschitz condition is replaced by one-sided Lipschitz condition, in order to guarantee the moment finiteness of numerical scheme, a modified multilevel Monte Carlo theta EM scheme is put forward and the second moment of two coupled paths is estimated.

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Source: https://tomesphere.com/paper/1904.07500