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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 $$ dX(t)=(aX(t)+f(X(t-r)))dt+\sigma X(t)dW(t),\quad t\ge 0$$ 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.

60H10Stochastic ordinary differential equations
34K50Stochastic functional-differential equations
93E03General theory of stochastic systems
Full Text: DOI
[1] Appleby, J.A.D., Buckwar, E., 2003. Noise induced oscillation in solutions of stochastic delay differential equations. Discussion Paper 9/2003, Sonderforschungsbereich 373, Humboldt-Universität zu Berlin. · Zbl 1105.34057
[2] Hobson, D. G.; Rogers, L. C. G.: Complete models with stochastic volatility. Math. finance 8, 27-48 (1998) · Zbl 0908.90012
[3] Karatzas, I.; Shreve, S. E.: Brownian motion and stochastic calculus. (1991) · Zbl 0734.60060
[4] Ladde, G. S.; Lakshmikantham, V.; Zhang, B. G.: Oscillation theory of differential equations with deviating arguments. (1987) · Zbl 0832.34071
[5] Mao, X.: Stochastic differential equations and their applications. (1997) · Zbl 0892.60057