Lack of strong completeness for stochastic flows

TL;DR

This paper constructs a 2D stochastic differential equation with bounded, smooth coefficients that is not strongly complete, disproving the long-standing open question about strong completeness under linear growth conditions.

Contribution

It provides the first explicit example of a bounded, smooth 2D SDE that fails to be strongly complete, answering a major open problem.

Findings

Constructed a bounded, smooth 2D SDE that is not strongly complete.

Disproved the assumption that linear growth implies strong completeness.

Answered a long-standing open question in stochastic analysis.

Abstract

It is well-known that a stochastic differential equation (SDE) on a Euclidean space driven by a Brownian motion with Lipschitz coefficients generates a stochastic flow of homeomorphisms. When the coefficients are only locally Lipschitz, then a maximal continuous flow still exists but explosion in finite time may occur. If -- in addition -- the coefficients grow at most linearly, then this flow has the property that for each fixed initial condition $x$, the solution exists for all times almost surely. If the exceptional set of measure zero can be chosen independently $x$, then the maximal flow is called {\em strongly complete}. The question, whether an SDE with locally Lipschitz continuous coefficients satisfying a linear growth condition is strongly complete was open for many years. In this paper, we construct a 2-dimensional SDE with coefficients which are even bounded (and smooth) and which is {\em not} strongly complete thus answering the question in the negative.