# Adaptive Euler-Maruyama method for SDEs with non-globally Lipschitz drift: Part II, infinite time interval

## TL;DR

This paper introduces an adaptive Euler-Maruyama method for ergodic SDEs with non-globally Lipschitz drift over infinite time, ensuring stability, uniform moment bounds, and strong convergence, enabling efficient multilevel Monte Carlo simulations.

## Contribution

It develops an adaptive timestep scheme for ergodic SDEs with non-globally Lipschitz drift, providing uniform bounds and convergence results over infinite time intervals.

## Key findings

- Numerical experiments confirm theoretical stability and convergence.
- Adaptive method achieves uniform moment bounds over infinite time.
- Supports efficient multilevel Monte Carlo simulations.

## Abstract

This paper proposes an adaptive timestep construction for an Euler-Maruyama approximation of the ergodic SDEs with a drift which is not globally Lipschitz over an infinite time interval. If the timestep is bounded appropriately, we show not only the stability of the numerical solution and the standard strong convergence order, but also that the bound for moments and strong error of the numerical solution are uniform in T, which allow us to introduce the adaptive multilevel Monte Carlo. Numerical experiments support our analysis.