On oscillations of the geometric Brownian motion with time-delayed drift (Q868267)

The authors consider the Ito stochastic differential equation \[ dX(t)=(aX(t)+f(X(t-r)))dt+\sigma X(t)dW(t),\quad t\geq 0 \] with scalar Brownian motion \(W\) and a locally bounded measurable function \(f\). Expressing the solution \(X\) in terms of the classical geometric Brownian motion, it can be proved that for a positive initial segment \((X(s),-r\leq s\leq 0)\) and non-negative \(f\), the process \(X\) remains positive a.s. On the other hand, the authors establish a condition on \(a\), \(\sigma\) and \(f\) such that the solution process with positive initial condition attains zero in finite time a.s. This condition is for instance satisfied if \(f\) is non-increasing with at least linear growth while \(a\) and \(\sigma\) are arbitrary.