Blow-up of a stable stochastic differential equation

2014/08/05 by Matti Leimbach, Leimbach, Matti, Michael Scheutzow +1
Computer Science · Economics, Econometrics and Finance · Engineering · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Probability (math.PR) #Stability and Controllability of Differential Equations #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1408.0933

openalex publication_date 2014/08/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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