Blow-up of a stable stochastic differential equation

TL;DR

This paper investigates a 2D stochastic differential equation that, despite having properties like invariance and completeness in the Markov sense, almost surely exhibits finite-time explosion for certain initial conditions, revealing nuanced behaviors of stochastic flows.

Contribution

It demonstrates that a stable SDE can have solutions that explode in finite time for some initial conditions, contrasting with its Markov process properties.

Findings

The SDE admits finite-time explosion for some initial conditions.

The associated stochastic flow is almost surely not strongly complete.

The Markov process has an invariant probability measure.

Abstract

We examine a 2-dimensional ODE which exhibits explosion in finite time. Considered as an SDE with additive white noise, it is known to be complete - in the sense that for each initial condition there is almost surely no explosion. Furthermore, the associated Markov process even admits an invariant probability measure. On the other hand, as we will show, the corresponding local stochastic flow will almost surely not be strongly complete, i.e.~there exist (random) initial conditions for which the solutions explode in finite time.