A note on the probabilistic stability of randomized Taylor schemes

TL;DR

This paper investigates the probabilistic stability properties of randomized Taylor schemes for ordinary differential equations, comparing them to deterministic schemes and analyzing three notions of stability.

Contribution

It introduces and analyzes probabilistic stability concepts for randomized Taylor schemes, providing fundamental properties and benchmarks against deterministic stability regions.

Findings

Probabilistic stability regions are characterized and compared to deterministic regions.

Fundamental properties of probabilistic stability are established.

Benchmarks against absolute stability regions are provided.

Abstract

We study the stability of randomized Taylor schemes for ODEs. We consider three notions of probabilistic stability: asymptotic stability, mean-square stability, and stability in probability. We prove fundamental properties of the probabilistic stability regions and benchmark them against the absolute stability regions for deterministic Taylor schemes.