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On oscillations of the geometric Brownian motion with time-delayed drift. (English) Zbl 1126.60048

The authors consider the Ito stochastic differential equation

dX(t)=(aX(t)+f(X(t-r)))dt+σX(t)dW(t),t0

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),-rs0) and non-negative f, the process X remains positive a.s. On the other hand, the authors establish a condition on a, σ 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 σ are arbitrary.

MSC:
60H10Stochastic ordinary differential equations
34K50Stochastic functional-differential equations
93E03General theory of stochastic systems