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Zbl 1126.60048
Gushchin, Alexander A.; Küchler, Uwe
On oscillations of the geometric Brownian motion with time-delayed drift. (English)
[J] Stat. Probab. Lett. 70, No. 1, 19-24 (2004). ISSN 0167-7152

The authors consider the Ito stochastic differential equation $$ dX(t)=(aX(t)+f(X(t-r)))dt+\sigma X(t)dW(t),\quad t\ge 0$$ 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 $(X(s),-r\le s\le 0)$ 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.
[Markus Reiss (Heidelberg)]

MSC 2000:: *60H10 Stochastic ordinary differential equations
34K50 Stochastic delay equations
93E03 General theory of stochastic systems

Keywords: geometric Brownian motion; stochastic delay differential equations

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